Puss button does not respond if a delay is used

you don't need delay.

just switch the contet of the display .

This switches between to LCD contents:

void updateLCD()
{
  const uint16_t interval = 60000;     // update Intervall in milliseconds
  static uint32_t previousMillis = 0;  // timestamp last update
  static byte state = 0;               // which "screen" is to display
  uint32_t currentMillis = millis();   

  if (currentMillis - previousMillis >= interval)
  {
    previousMillis = currentMillis;
    switch (state)
    {
      case 0:                          // show screen 0
        Pressure = bme280.getPressure();
        lcd.clear();
        lcd.print("BME280 pressure:");
        lcd.setCursor(0, 1);
        lcd.print(Pressure);
        state = 1; // show screen 1 next time
        break;
      case 1:                          // show screen 1
        Temp = sht.getTemperature();
        lcd.clear();
        lcd.print("SHT3x temp:");
        lcd.setCursor(0, 1);
        lcd.print(Temp);
        state = 0;
        break;
    }
  }
}

and call

updateLCD();

in your loop().