LED show start with button

All variables used with millis() or micros() timing should be unsigned long data type. I did not fix them, I leave that to you.

Use the state change detection method to sense when the button becomes pressed.

Does this do what you want? Added code marked with // *********.

unsigned int intervals_1[] = {1425, 450, 375, 250, 325, 250, 300, 250, 325, 250, 300, 250, 275, 250, 300, 250, 325, 250, 250, 250, 350, 250, 300, 250, 325, 250, 300, 250, 325, 250, 275, 250, 300, 250, 300, 250, 325, 250, 350, 250, 300, 250, 275, 250, 300, 250, 300, 250, 300, 250, 325, 250, 275, 250, 325, 250, 300, 250, 300, 250, 325, 250, 300, 250, 300, 250, 325, 250, 275, 250, 300, 250, 150, 250, 275, 250, 300, 250, 300, 250, 300, 250, 325, 250, 275, 250, 325, 250, 300, 250, 300, 250, 325, 250, 300, 250, 300, 250, 325, 250, 275, 250, 300, 250, 400, 250, 275, 250, 300, 250, 300, 250, 300, 250, 325, 250, 275, 250, 325, 250, 300, 250, 300, 250, 325, 250, 300, 250, 300, 250, 350, 250, 400, 250, 325, 250, 325, 250, 300, 250, 325, 250, 600, 1075, 575, 1675, 600, 1675, 600, 1775, 700, 1250, 575, 1675, 600, 1675, 600, 1775, 700, 175, 50, 125, 975, 125, 1025, 125, 1025, 125, 950, 125, 975, 125, 1025, 125, 1025, 125, 975, 125, 975, 125, 1025, 125, 1025, 125, 1025, 125, 975, 125, 1025, 125, 1025, 125, 950, 125, 975, 125, 1025, 125, 1025, 125, 975, 125, 975, 125, 1025, 125, 1375, 250, 1400, 275};
unsigned int intervals_2[] = {1675, 200, 375, 250, 325, 250, 300, 250, 6950, 250, 300, 250, 325, 250, 350, 250, 2475, 250, 325, 250, 3625, 250, 325, 250, 275, 250, 300, 250, 2325, 250, 325, 250, 3625, 250, 325, 250, 275, 250, 300, 250, 2575, 250, 325, 250, 3625, 250, 5200, 575, 1675, 575, 1725, 575, 1775, 100, 500, 100, 1250, 575, 1675, 575, 1725, 575, 1775, 100, 500, 100, 1125, 100, 1000, 100, 1050, 100, 1050, 100, 975, 100, 1000, 100, 1050, 100, 1050, 100, 1000, 100, 1000, 100, 1050, 100, 1050, 100, 1050, 100, 1000, 100, 1050, 100, 1050, 100, 975, 100, 1000, 100, 1050, 100, 1050, 100, 1000, 100, 1000, 100, 1900, 300, 750, 225};
unsigned int intervals_3[] = {3950, 250, 300, 250, 1400, 250, 250, 250, 1475, 250, 300, 250, 3675, 250, 275, 250, 3625, 250, 275, 250, 3500, 250, 275, 250, 3625, 250, 275, 250, 3750, 250, 275, 250, 3625, 250, 275, 250, 5825, 1075, 575, 1675, 600, 1675, 600, 1675, 200, 100, 300, 100, 275, 1075, 575, 1675, 600, 1675, 600, 1675, 200, 100, 300, 100, 675, 100, 350, 100, 550, 100, 350, 100, 600, 100, 350, 100, 600, 100, 350, 100, 525, 100, 350, 100, 550, 100, 350, 100, 600, 100, 350, 100, 600, 100, 350, 100, 550, 100, 350, 100, 550, 100, 350, 100, 600, 100, 350, 100, 600, 100, 350, 100, 600, 100, 350, 100, 550, 100, 350, 100, 600, 100, 350, 100, 600, 100, 350, 100, 525, 100, 350, 100, 550, 100, 350, 100, 600, 100, 350, 100, 600, 100, 350, 100, 550, 100, 350, 100, 550, 100, 350, 100, 600, 100, 800, 250, 850, 400};
unsigned int intervals_4[] = {5025, 250, 300, 250, 3675, 250, 275, 250, 3600, 250, 300, 250, 3700, 250, 300, 250, 3375, 250, 300, 250, 3700, 250, 300, 250, 3625, 250, 300, 250, 3700, 250, 300, 250, 5750, 575, 1675, 575, 1725, 575, 1975, 100, 100, 100, 1450, 575, 1675, 575, 1725, 575, 1975, 100, 100, 100, 26175, 250, 2350, 525};
unsigned int intervals_5[] = {7250, 250, 300, 250, 8050, 250, 325, 250, 7850, 250, 325, 250, 8100, 250, 325, 250, 6925, 1075, 575, 1675, 600, 1675, 600, 1675, 400, 100, 475, 1075, 575, 1675, 600, 1675, 600, 1675, 400, 100, 1125, 75, 1025, 75, 1075, 75, 1075, 75, 1000, 75, 1025, 75, 1075, 75, 1075, 75, 1025, 75, 1025, 75, 1075, 75, 1075, 75, 1075, 75, 1025, 75, 1075, 75, 1075, 75, 1000, 75, 1025, 75, 1075, 75, 1075, 75, 1025, 75, 1025, 75, 1075, 75, 75, 250, 3150, 900};
unsigned int intervals_6[] = {3800, 1275, 7425, 1275, 23175, 1275, 4300, 575, 1675, 575, 1725, 575, 3725, 575, 1675, 575, 1725, 575, 15700, 3200, 6925, 2375};
unsigned int intervals_7[] = {21400, 150, 250, 150, 275, 150, 1250, 150, 250, 150, 275, 150, 1250, 150, 250, 150, 275, 150, 1175, 150, 300, 150, 300, 150, 1250, 150, 250, 150, 275, 150, 1250, 150, 250, 150, 275, 150, 1250, 150, 250, 150, 275, 150, 6000, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 17925, 5200};
unsigned int intervals_8[] = {20275, 150, 150, 150, 125, 150, 125, 150, 7175, 150, 150, 150, 125, 150, 125, 150, 125, 150, 125, 150, 150, 150, 11150, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 9375, 4050}; // zatím hotovo
unsigned int intervals_9[] = {41450, 550, 275, 250, 500, 1050, 475, 250, 500, 1050, 400, 250, 500, 300, 2525, 550, 275, 250, 500, 1050, 475, 250, 500, 1050, 400, 250, 500, 300, 31425, 2900};
unsigned int intervals_10[] = {41450, 550, 525, 250, 250, 1050, 725, 250, 250, 1050, 650, 250, 250, 650, 2175, 550, 525, 250, 250, 1050, 725, 250, 250, 1050, 650, 250, 250, 650, 31075, 2900};
unsigned int intervals_11[] = {41450, 550, 775, 1300, 975, 1300, 900, 550, 350, 1825, 350, 550, 775, 1300, 975, 1300, 900, 550, 350, 1825, 29250, 2900};


const byte NUM_OF_INTERVALS_1 = sizeof(intervals_1) / sizeof(unsigned int);
const byte NUM_OF_INTERVALS_2 = sizeof(intervals_2) / sizeof(unsigned int);
const byte NUM_OF_INTERVALS_3 = sizeof(intervals_3) / sizeof(unsigned int);
const byte NUM_OF_INTERVALS_4 = sizeof(intervals_4) / sizeof(unsigned int);
const byte NUM_OF_INTERVALS_5 = sizeof(intervals_5) / sizeof(unsigned int);
const byte NUM_OF_INTERVALS_6 = sizeof(intervals_6) / sizeof(unsigned int);
const byte NUM_OF_INTERVALS_7 = sizeof(intervals_7) / sizeof(unsigned int);
const byte NUM_OF_INTERVALS_8 = sizeof(intervals_8) / sizeof(unsigned int);
const byte NUM_OF_INTERVALS_9 = sizeof(intervals_9) / sizeof(unsigned int);
const byte NUM_OF_INTERVALS_10 = sizeof(intervals_10) / sizeof(unsigned int);
const byte NUM_OF_INTERVALS_11 = sizeof(intervals_11) / sizeof(unsigned int);

byte currentInterval_1 = 0;
byte currentInterval_2 = 0;
byte currentInterval_3 = 0;
byte currentInterval_4 = 0;
byte currentInterval_5 = 0;
byte currentInterval_6 = 0;
byte currentInterval_7 = 0;
byte currentInterval_8 = 0;
byte currentInterval_9 = 0;
byte currentInterval_10 = 0;
byte currentInterval_11 = 0;

const byte LEDtoBlink_1 = 22;
const byte LEDtoBlink_2 = 23;
const byte LEDtoBlink_3 = 24;
const byte LEDtoBlink_4 = 25;
const byte LEDtoBlink_5 = 26;
const byte LEDtoBlink_6 = 27;
const byte LEDtoBlink_7 = 28;
const byte LEDtoBlink_8 = 29;
const byte LEDtoBlink_9 = 30;
const byte LEDtoBlink_10 = 31;
const byte LEDtoBlink_11 = 32;



unsigned int previousMillis_1 = 0;
unsigned int previousMillis_2 = 0;
unsigned int previousMillis_3 = 0;
unsigned int previousMillis_4 = 0;
unsigned int previousMillis_5 = 0;
unsigned int previousMillis_6 = 0;
unsigned int previousMillis_7 = 0;
unsigned int previousMillis_8 = 0;
unsigned int previousMillis_9 = 0;
unsigned int previousMillis_10 = 0;
unsigned int previousMillis_11 = 0;

const byte buttonPin = 33;  // ****** button to pin 33 and ground ********

void setup()
{
   Serial.begin(115200);

   pinMode(buttonPin, INPUT_PULLUP); // *********************

   pinMode(LEDtoBlink_1, OUTPUT);
   pinMode(LEDtoBlink_2, OUTPUT);
   pinMode(LEDtoBlink_3, OUTPUT);
   pinMode(LEDtoBlink_4, OUTPUT);
   pinMode(LEDtoBlink_5, OUTPUT);
   pinMode(LEDtoBlink_6, OUTPUT);
   pinMode(LEDtoBlink_7, OUTPUT);
   pinMode(LEDtoBlink_8, OUTPUT);
   pinMode(LEDtoBlink_9, OUTPUT);
   pinMode(LEDtoBlink_10, OUTPUT);
   pinMode(LEDtoBlink_11, OUTPUT);
}

void loop()
{
   static bool buttonStateLast = HIGH; // *************************
   bool buttonState = digitalRead(buttonPin); // **********************
   if (buttonState != buttonStateLast) // **************************
   {
      if (buttonState == LOW)  // *********************
      {
         unsigned int currentMillis_1 = millis();
         unsigned int currentMillis_2 = millis();
         unsigned int currentMillis_3 = millis();
         unsigned int currentMillis_4 = millis();
         unsigned int currentMillis_5 = millis();
         unsigned int currentMillis_6 = millis();
         unsigned int currentMillis_7 = millis();
         unsigned int currentMillis_8 = millis();
         unsigned int currentMillis_9 = millis();
         unsigned int currentMillis_10 = millis();
         unsigned int currentMillis_11 = millis();
         unsigned int currentMillis_x = millis();


         if (currentMillis_1 - previousMillis_1 >= intervals_1[currentInterval_1])  //if the time interval has elapsed   && btn_st_1 == HIGH
         {
            currentInterval_1 = currentInterval_1 + 1;     // select the next interval in the list
            if (currentInterval_1 >= NUM_OF_INTERVALS_1)
               digitalWrite(LEDtoBlink_1, HIGH);
            digitalWrite(LEDtoBlink_1, not digitalRead(LEDtoBlink_1));    //change state of the LED
            previousMillis_1 = currentMillis_1;                //save the time of change
         }

         if (currentMillis_2 - previousMillis_2 >= intervals_2[currentInterval_2])  //if the time interval has elapsed
         {
            currentInterval_2 = currentInterval_2 + 1;     // select the next interval in the list
            if (currentInterval_2 >= NUM_OF_INTERVALS_2)
               digitalWrite(LEDtoBlink_2, HIGH);
            digitalWrite(LEDtoBlink_2, not digitalRead(LEDtoBlink_2));    //change state of the LED
            previousMillis_2 = currentMillis_2;                //save the time of change
         }

         if (currentMillis_3 - previousMillis_3 >= intervals_3[currentInterval_3])  //if the time interval has elapsed
         {
            currentInterval_3 = currentInterval_3 + 1;     // select the next interval in the list
            if (currentInterval_3 >= NUM_OF_INTERVALS_3)
               digitalWrite(LEDtoBlink_3, HIGH);
            digitalWrite(LEDtoBlink_3, not digitalRead(LEDtoBlink_3));    //change state of the LED
            previousMillis_3 = currentMillis_3;                //save the time of change
         }


         if (currentMillis_4 - previousMillis_4 >= intervals_4[currentInterval_4])  //if the time interval has elapsed
         {
            currentInterval_4 = currentInterval_4 + 1;     // select the next interval in the list
            if (currentInterval_4 >= NUM_OF_INTERVALS_4)
               digitalWrite(LEDtoBlink_4, HIGH);
            digitalWrite(LEDtoBlink_4, not digitalRead(LEDtoBlink_4));    //change state of the LED
            previousMillis_4 = currentMillis_4;                //save the time of change
         }


         if (currentMillis_5 - previousMillis_5 >= intervals_5[currentInterval_5])  //if the time interval has elapsed
         {
            currentInterval_5 = currentInterval_5 + 1;     // select the next interval in the list
            if (currentInterval_5 >= NUM_OF_INTERVALS_5)
               digitalWrite(LEDtoBlink_5, HIGH);
            digitalWrite(LEDtoBlink_5, not digitalRead(LEDtoBlink_5));    //change state of the LED
            previousMillis_5 = currentMillis_5;                //save the time of change_6
         }


         if (currentMillis_6 - previousMillis_6 >= intervals_6[currentInterval_6])  //if the time interval has elapsed
         {
            currentInterval_6 = currentInterval_6 + 1;     // select the next interval in the list
            if (currentInterval_6 >= NUM_OF_INTERVALS_6)
               digitalWrite(LEDtoBlink_6, HIGH);
            digitalWrite(LEDtoBlink_6, not digitalRead(LEDtoBlink_6));    //change state of the LED
            previousMillis_6 = currentMillis_6;                //save the time of change
         }

         if (currentMillis_7 - previousMillis_7 >= intervals_7[currentInterval_7])  //if the time interval has elapsed
         {
            currentInterval_7 = currentInterval_7 + 1;     // select the next interval in the list
            if (currentInterval_7 >= NUM_OF_INTERVALS_7)
               digitalWrite(LEDtoBlink_7, HIGH);
            digitalWrite(LEDtoBlink_7, not digitalRead(LEDtoBlink_7));    //change state of the LED
            previousMillis_7 = currentMillis_7;                //save the time of change
         }

         if (currentMillis_8 - previousMillis_8 >= intervals_8[currentInterval_8])  //if the time interval has elapsed
         {
            currentInterval_8 = currentInterval_8 + 1;     // select the next interval in the list
            if (currentInterval_8 >= NUM_OF_INTERVALS_8)
               digitalWrite(LEDtoBlink_8, HIGH);
            digitalWrite(LEDtoBlink_8, not digitalRead(LEDtoBlink_8));    //change state of the LED
            previousMillis_8 = currentMillis_8;                //save the time of change
         }

         if (currentMillis_9 - previousMillis_9 >= intervals_9[currentInterval_9])  //if the time interval has elapsed
         {
            currentInterval_9 = currentInterval_9 + 1;     // select the next interval in the list
            if (currentInterval_9 >= NUM_OF_INTERVALS_9)
               digitalWrite(LEDtoBlink_9, HIGH);
            digitalWrite(LEDtoBlink_9, not digitalRead(LEDtoBlink_9));    //change state of the LED
            previousMillis_9 = currentMillis_9;                //save the time of change
         }

         if (currentMillis_10 - previousMillis_10 >= intervals_10[currentInterval_10])  //if the time interval has elapsed
         {
            currentInterval_10 = currentInterval_10 + 1;     // select the next interval in the list
            if (currentInterval_10 >= NUM_OF_INTERVALS_10)
               digitalWrite(LEDtoBlink_10, HIGH);
            digitalWrite(LEDtoBlink_10, not digitalRead(LEDtoBlink_10));    //change state of the LED
            previousMillis_10 = currentMillis_10;                //save the time of change
         }

         if (currentMillis_11 - previousMillis_11 >= intervals_11[currentInterval_11])  //if the time interval has elapsed
         {
            currentInterval_11 = currentInterval_11 + 1;     // select the next interval in the list
            if (currentInterval_11 >= NUM_OF_INTERVALS_11)
               digitalWrite(LEDtoBlink_11, HIGH);
            digitalWrite(LEDtoBlink_11, not digitalRead(LEDtoBlink_11));    //change state of the LED
            previousMillis_11 = currentMillis_11;                //save the time of change
         }
         buttonStateLast = buttonState; //**********************
      }
   }
}