Switching on the diode for some time, as well as execution of the next program in Arduino

The circuit made in tinkercad
I use the relay, because I have only 4 wires to led diode and this two switches.

int led = 12; // red led
int s1 = 9; //switch 1
int s2 = 10; //switch 1
int k1 = 3; // first blue led
int k2 = 2; // second blue led
int y1 = 11; // relay
unsigned long startTime1 = 0;
unsigned long startTime2 = 0;
const int led1Duration = 6000; // first blue led time
const int led2Duration = 12000; // second blue led time
void setup()
{
pinMode(led, OUTPUT);
pinMode(k1, OUTPUT);
pinMode(k2, OUTPUT);
pinMode(y1, OUTPUT);
pinMode(s1, INPUT);
pinMode(s2, INPUT);
}
void loop()
{
if (digitalRead(s1) == HIGH and digitalRead(s2) == LOW)
{	
digitalWrite(k1, HIGH);
digitalWrite(y1, HIGH);
startTime1 = millis(); 
}
else
{
digitalWrite(led, LOW);
}
if (digitalRead(s2) == HIGH and digitalRead(s1) == LOW)
{
digitalWrite(k2, HIGH);
digitalWrite(y1, HIGH);
startTime2 = millis(); 
}
else
{
digitalWrite(led, LOW);
}
if (digitalRead(k1) == HIGH && (millis() - startTime1 >= led1Duration)) 
{
digitalWrite(k1, LOW);
digitalWrite(y1, LOW);
}
if (digitalRead(k2) == HIGH && (millis() - startTime2 >= led2Duration)) 
{
digitalWrite(k2, LOW);
digitalWrite(y1, LOW);
}
if (digitalRead(k1) == HIGH and digitalRead(k2) == LOW)
{
digitalWrite(led, HIGH);
delay(200);
digitalWrite(led, LOW);
delay(200);
}
if (digitalRead(k2) == HIGH and digitalRead(k1) == LOW)
{
digitalWrite(led, HIGH);
delay(500);
digitalWrite(led, LOW);
delay(500);
}
if (digitalRead(k2) == HIGH and digitalRead(k1) == HIGH)
{
digitalWrite(led, HIGH);
}
}

Is there easier way to do that, cause when I assembled it the switches work very slow that blue led turn on after some time. Next I use the esp2866 but the relay for 3V doesn't turn on. How can I make it to work?

@Peter is true : avoid blocking delay code.

here is an example : I have a little simplified your code and added else if to make it more readable.

update1 :

   int led = 12; // red led
int s1 = 9; //switch 1
int s2 = 10; //switch 1
int k1 = 3; // first blue led
int k2 = 2; // second blue led
int y1 = 11; // relay
unsigned long startTime1 = 0;
unsigned long startTime2 = 0;
unsigned long t200 = 200;
unsigned long t500 = 500;
const int led1Duration = 6000; // first blue led time
const int led2Duration = 12000; // second blue led time
void setup()
{
pinMode(led, OUTPUT);
pinMode(k1, OUTPUT);
pinMode(k2, OUTPUT);
pinMode(y1, OUTPUT);
pinMode(s1, INPUT);
pinMode(s2, INPUT);
}
void loop()
{
if (digitalRead(s1) && !digitalRead(s2)) {
digitalWrite(k1, HIGH);
digitalWrite(y1, HIGH);
startTime1 = millis();
blinkLed200();
} else if (!digitalRead(s1) && digitalRead(s2)) {
digitalWrite(k2, HIGH);
digitalWrite(y1, HIGH);
startTime2 = millis();
blinkLed500();
}
if (digitalRead(k1) && (millis() - startTime1 >= led1Duration)) {
digitalWrite(k1, LOW);
digitalWrite(y1, LOW);
digitalWrite(led, LOW);
}
if (digitalRead(k2) && (millis() - startTime2 >= led2Duration)) {
digitalWrite(k2, LOW);
digitalWrite(y1, LOW);
digitalWrite(led, LOW);
}
}
void blinkLed200() {
if (millis() - t200 > 200) { //tick every 200mS
t200 = millis();
digitalWrite(led, !digitalRead(led));
}
}
void blinkLed500() {
if (millis() - t500 > 500) { //tick every 500mS
t500 = millis();
digitalWrite(led, !digitalRead(led));
}
}

The 3V relays have a very low resistance and I am not sure that the digital outputs are powerful enough. Personally I always switch relays via a transistor.