Frequency of loop, actualTime - storedTime

not really sure what the specific problem is. i simulated your code and cleaned up some things.

the button processing: made INPUT_PULLUP, moved button1_last outside of loop() and simplified the logic (no need for "Bounce").

i moved reading millis() to the top of loop()

you include Stepper, but don't use it's functions. those sequences seem redundant

consider

#define MyHW
#ifdef MyHW
class Stepper {
  public:
    Stepper (int z, int a, int b, int c, int d)  {
        pinMode (a, OUTPUT);
        pinMode (b, OUTPUT);
        pinMode (c, OUTPUT);
        pinMode (d, OUTPUT);
    };

    void step (int a)   { };
};

const int Step1Pin=10;
const int Step2Pin=11;
const int Step3Pin=12;
const int Step4Pin=13;

class LiquidCrystal {
  public:
    LiquidCrystal (int rs, int en, int d4, int d5, int d6, int d7)  { };

    void begin   (int a, int b)   { };
    void clear   (void)           { };
#if 0
    void print   (const char *s)  { Serial.print (s); };
    void println (const char *s)  { Serial.println (s); };
#else
    void print   (const char *s)  { };
    void println (const char *s)  { };
#endif
};

class Bounce {
  public:
    Bounce ()   { };
    void attach   (int a)  { };
    void interval (int a)  { };
};

byte pot        = A0;
byte buttonPin  = A1;
byte buttonPin2 = A2;

// ---------------------------------------------------------
#else
#include <LiquidCrystal.h>
#include <Bounce2.h>
#include <Stepper.h>

const int Step1Pin=6;
const int Step2Pin=7;
const int Step3Pin=8;
const int Step4Pin=9;

int pot=A1;
int buttonPin=10;
int buttonPin2=13;
#endif

char s [80];

// -----------------------------------------------------------------------------
const int stepsPerRevolution = 200; // change this to fit the number of steps per revolution
// for your motor
// initialize the stepper library on pins 8 through 11:
Stepper myStepper(stepsPerRevolution, Step1Pin, Step2Pin, Step3Pin, Step4Pin);
int stepCount = 0; // number of steps the motor has taken

// initialize the library by associating any needed LCD interface pin
// with the arduino pin number it is connected to
const int rs = 12, en = 11, d4 = 5, d5 = 4, d6 = 3, d7 = 2;
int buttonNew;
int buttonOld=0;
int LCDState=0;
int dt=500;
int controloption;
int count=0;
unsigned long predkosc;

unsigned long actualTime = 0;
unsigned long storedTime=0;
int stepNo=0;
int czujnik=0;
LiquidCrystal lcd(rs, en, d4, d5, d6, d7);
Bounce debouncer = Bounce();

void setup() {
    Serial.begin(9600);

    pinMode(buttonPin,  INPUT_PULLUP);
    pinMode(buttonPin2, INPUT_PULLUP);
    // Po konfiguracji przycisku, ustawienie dziaania funkcji Bounce :
    debouncer.attach(buttonPin);
    debouncer.interval(25); // interwa w ms
    // set up the LCD's number of columns and rows:
    lcd.begin(16, 2);
    // Print a message to the LCD.
}

int but1_last = HIGH;

void loop()
{
    actualTime = millis();
    czujnik  = analogRead(pot);
    predkosc = map(czujnik, 1, 1000, -10, 1000);
 // Serial.println(predkosc);

    int but1          = digitalRead(buttonPin);
    int button2state  = digitalRead(buttonPin2);

#if 0
    if (button1_state != button1_last){//debouncer2
        delay(50);
        button1_state = digitalRead(buttonPin);
    }

    while (button1_state != button1_last)
    {
        count++;
        storedTime=actualTime;
        button1_last = button1_state;
        delay(50);
        if (count >= 5) {
            count = 1;
        }
    }
#else
    if (but1_last != but1)  {
        but1_last = but1;
        delay (10);         // debounce

        if (LOW == but1)  {
            storedTime = actualTime;
            count = (count+1) % 6;
        }
    }
#endif

    sprintf (s, "cnt %3d, stepNo %3d, %8ld, %8ld, %8ld",
        count, stepNo, actualTime, storedTime, predkosc);
    Serial.println (s);

    switch (count) {
    case 1:
        lcd.print("sterowanie");
        lcd.println("pelnokrokowe");
        if ( (actualTime - storedTime) >= predkosc) {
            if (stepNo == 1) {
                digitalWrite(Step1Pin, HIGH);
                digitalWrite(Step2Pin, LOW);
                digitalWrite(Step3Pin, LOW);
                digitalWrite(Step4Pin, HIGH);
            }
            if (stepNo == 2) {
                digitalWrite(Step1Pin, HIGH);
                digitalWrite(Step2Pin, LOW);
                digitalWrite(Step3Pin, HIGH);
                digitalWrite(Step4Pin, LOW);
            }
            if (stepNo == 3) {
                digitalWrite(Step1Pin, LOW);
                digitalWrite(Step2Pin, HIGH);
                digitalWrite(Step3Pin, HIGH);
                digitalWrite(Step4Pin, LOW);
            }
            if (stepNo == 4) {
                digitalWrite(Step1Pin, LOW);
                digitalWrite(Step2Pin, HIGH);
                digitalWrite(Step3Pin, LOW);
                digitalWrite(Step4Pin, HIGH);
                stepNo = 0;
            }
            stepNo++;
            storedTime = actualTime;
        }
        if (button2state==0){
            lcd.clear();
            lcd.print("zalaczono funkcje");
            delay(1000);
        }
        break;

    case 2:
        lcd.clear();
        lcd.print("sterowanie");
        lcd.println("falowe");
        if (actualTime - storedTime >= predkosc) {
            if (stepNo == 1) {
                digitalWrite(Step1Pin, LOW); //B+
                digitalWrite(Step2Pin, LOW);//B-
                digitalWrite(Step3Pin, HIGH);//A+
                digitalWrite(Step4Pin, LOW);//A-
            }
            if (stepNo == 2) {
                digitalWrite(Step1Pin, HIGH);
                digitalWrite(Step2Pin, LOW);
                digitalWrite(Step3Pin, LOW);
                digitalWrite(Step4Pin, LOW);
            }
            if (stepNo == 3) {
                digitalWrite(Step1Pin, LOW);
                digitalWrite(Step2Pin, LOW);
                digitalWrite(Step3Pin, LOW);
                digitalWrite(Step4Pin, HIGH);
            }
            if (stepNo == 4) {
                digitalWrite(Step1Pin, LOW);
                digitalWrite(Step2Pin, HIGH);
                digitalWrite(Step3Pin, LOW);
                digitalWrite(Step4Pin, LOW);
                stepNo = 0;
            }
            stepNo++;
            storedTime = actualTime;
        }
        break;

    case 3:
        lcd.clear();
        lcd.print("silnik prawo");
        if (button2state==0){
            lcd.clear();
            lcd.print("zalaczono funkcje");
            //delay(1);
            myStepper.step(1);
            stepCount++;
        }
        break;

    case 4:
        lcd.clear();
        lcd.print("sterowanie");
        lcd.println("polkrokowe");
        if (actualTime - storedTime >= predkosc)
        {
            if (stepNo == 1) {              // A+B+  A-B+  A-B-  A+B-
                digitalWrite(Step1Pin, LOW); //B+           A+0  A+B+  0B+  A-B+  A-0  A-B- 0B- A+B-
                digitalWrite(Step2Pin, LOW); //B-
                digitalWrite(Step3Pin, LOW);// A-
                digitalWrite(Step4Pin, HIGH); //A+
            }
            if (stepNo == 2) {
                digitalWrite(Step1Pin, HIGH);
                digitalWrite(Step2Pin, LOW);
                digitalWrite(Step3Pin, LOW);
                digitalWrite(Step4Pin, HIGH);
            }
            if (stepNo == 3) {
                digitalWrite(Step1Pin, HIGH);
                digitalWrite(Step2Pin, LOW);
                digitalWrite(Step3Pin, LOW);
                digitalWrite(Step4Pin, LOW);
            }
            if (stepNo == 4) {
                digitalWrite(Step1Pin, HIGH);
                digitalWrite(Step2Pin, LOW);
                digitalWrite(Step3Pin, HIGH);
                digitalWrite(Step4Pin, LOW);
            }
            if (stepNo == 5) {
                digitalWrite(Step1Pin, LOW);
                digitalWrite(Step2Pin, LOW);
                digitalWrite(Step3Pin, HIGH);
                digitalWrite(Step4Pin, LOW);
            }
            if (stepNo == 6) {
                digitalWrite(Step1Pin, LOW);
                digitalWrite(Step2Pin, HIGH);
                digitalWrite(Step3Pin, HIGH);
                digitalWrite(Step4Pin, LOW);
            }
            if (stepNo == 7) {
                digitalWrite(Step1Pin, LOW);
                digitalWrite(Step2Pin, HIGH);
                digitalWrite(Step3Pin, LOW);
                digitalWrite(Step4Pin, LOW);
            }
            if (stepNo == 8) {
                digitalWrite(Step1Pin, LOW);
                digitalWrite(Step2Pin, HIGH);
                digitalWrite(Step3Pin, LOW);
                digitalWrite(Step4Pin, HIGH);
                stepNo = 0;
            }
            stepNo++;
            storedTime = actualTime;
        }
        break;
    }
}