Yes the relay gives a click sound on 12 v supply I have cross check led, there is no fault with led. I get 0v between NC and common

Isn't that what you would expect? NC is not connected to anything.

 

Here's my take on the pictures and schematics. Keep in mind part of this is a guess. Others may likely have a better idea what's going on. Also, NO's and NC's may be reversed but it looks fairly confident the common (C) is correct. But then again, maybe not.

 

Yes the relay gives a click sound on 12 v supply I have cross check led, there is no fault with led. I get 0v between NC and common

Yes... you expected differently? NC is a closed switch. A wire until it’s opened. Voltage across a wire is 0V.

 

Relay datasheet https://pdf.directindustry.com/pdf/o-e-n-india-ltd/series-57/63536-147352.html

May be I am not identifying correct NO, NC and common pin of Relay

This is my wiring connection for NO, NC and common pins

 

Based on the drawing you post in #24, why do you expect a voltage between NC and COM?

Assuming the positive terminal of the 5V DC is connected to NO and its negative pole to the same ground to which the LED is connected, then:
1) There should be voltage between NC and NO when the relay is NOT energized.
2) There should be voltage between COM and NO when the relay is NOT energized.
3) When the relay is energized, there should be no voltage (or very low due to contact resistance) between NO and COM.
4) When the relay is energized, there should be no voltage between NC and anything. NC is floating.