You can use the watchdog timer, but you have to change the way you measure the elapsed time. Try this code:

#include <avr/wdt.h>

unsigned long myTime; // Millis() function time value
unsigned long elapsedTime;
unsigned long timeExpected = 5000; // I want to reset it in every 5 secs for testing.
const int RESET_PIN = 2; // Reset pin's connection

void setup() {
  Serial.begin(9600);
  pinMode(LED_BUILTIN, OUTPUT);
  myTime = millis();
  Serial.println("INIT");
  wdt_enable(WDTO_2S); //Enable wdt every two seconds

}
void loop() {

  //************RESET HANDLER********************//
  elapsedTime = millis() - myTime;
  if (elapsedTime > timeExpected) {
    Serial.print("RESET");
    while (1); //Intentionally halt the program, so the watchdog performs the reset    
  }
  //**********************************************
  //****************REST OF THE PROGRAM***********
  wdt_reset(); //Ensure one call to this function at least one time in two seconds or less
  digitalWrite(LED_BUILTIN, HIGH);
  Serial.println(elapsedTime); // prints time since program started
  delay(1000);           
}

Please note that if you have multiple delays in your program, you have to ensure the call to wdt_reset() in order to prevent unwanted resets. Personally I try not to use delays and would recommend avoid delays as much as possible. You may check out this tutorial if you want to see how to avoid delays. If you achieve avoiding delays you can call wdt_reset() just one time at the begining of the loop.

You sent them to 0 then when the loop starts again you reset them to millis()/1000 & millis()/60000. Don't reset them at the beginning of the loop.

Here is the circuit I use which depends on a heartbeat from the arduino and resets the arduino if 8 pulses are missed and also enables the arduino code to know that an external watchdog reset occured

and the Arduino code which can be incorporated into your sketch

//Watchdog
#define ResetDetect 8       // watchdog detect pin, HIGH if a watchdog reset has occured
#define heartbeat 9         // heartbeat pin
int pulseState = LOW;       // pulseState used to set the heartbeat pin
long lastbeat = 0;          // will store last time the heartbeat pin was updated
long HeartBeatFreq = 500;   // interval at which to blink (milliseconds)
boolean ResetHappened;      // Set to true if a watchdog reset has occurred

void setup(){
  pinMode(ResetDetect, INPUT);    // set Watchdog Reset sensing pin as input
  digitalWrite(ResetDetect, HIGH);// and turn on pullup  
  pinMode(heartbeat, OUTPUT);     // set the heartbeat pin as output:

    // Check if Restarting after Watchdog Reset
    // NB must come before heartbeat resets external counter
    int ResetSet = digitalRead(ResetDetect);
    if (ResetSet == HIGH){
      ResetHappened = true;
    }
    else {
      ResetHappened = false;
    }
}

void loop(){

  // Heartbeat resets external watchdog when pin goes high
  if ((long)( millis() - (lastbeat + HeartBeatFreq)) >= 0) {    
    lastbeat = millis();    
    // if the LED is off turn it on and vice-versa:
    if (pulseState == LOW)
      pulseState = HIGH;
    else
      pulseState = LOW;
    // set the LED with the ledState of the variable:
    digitalWrite(heartbeat, pulseState);
  }
}

If there are long running sections of code the 'heartbeat' could be put into a function and that function called in the appropriate places to avoid unintended timeouts/resets