13 hours ! Forget the lounge and get a room in the airside transit hotel instead.
Your baggage should be checked through at Heathrow, confirm with the counter staff. You may get the boarding pass for the next leg or you may need to go to the transit desk on arrival in Delhi.
This time of the year is fog season so be prepared for disruption in Delhi, particularly with an early morning arrival.
Air India wouldn't be my first choice on this route and nor would the Gulf airlines (in economy) as they carry migrant workers who may not be the best of travellers.
It may be an experience for you.
