I have a layover of 15 hours at LHR. Arriving at 6 pm in the evening by BA from Mumbai and departing next day at 9 am to LAX.
Does BA provide overnight hotel accommodation and what is the rate if to stay on own?
BA would only provide accommodation if you're flying business or first class and/or agreed this/included into the ticket when you bought it. There are a lot of hotels in and around Heathrow airport though so you should have no difficulties booking a room yourself.
One thing you need to be careful about though is your visa requirements. If you need a visa to visit UK, then you are more than likely to require a transit visa to exit the airport terminal, as all hotels are outside. Provided you can get out of the terminal, the hotels in the airport are not the cheapest around (as usual) - they start at about £100 per night for one person and go up and up.
Yet, with that much time on your hands, you can easily get farther out from the airport and get a cheaper hotel. A few stops on the underground - and you can get a hotel for £60-70. If you're willing to settle for something less, then you can get a hostel for £15 or so a night.
A long time ago (so actually one of BA's predecessors, BOAC) did own hotels and accommodate passengers within them during long layovers - all as part of the flight ticket. However that ceased many decades ago, except perhaps I think they may still have a facility just for their own staff, near Heathrow.
Ibis is also near Heathrow, on the Bath Road it is almost inside the airport perimeter. I mention this because a friend stayed there last week for £40. It is currently advertising "From £51", so if you want to spent £100/night you should be able to obtain quite high quality in the area.
The airport Hoppa Bus runs every 20-30 minutes costing £4.50; take bus H6 for Terminals 2&3 or H56 for Terminals 4&5. There are free local buses from Central Bus Station to Harlington Corner (Buses 105,111,140).
Heathrow is far enough away from the centre that the exorbitant hotel prices there have limited influence on the collection of hotels that are close to Heathrow but not to anywhere else of any great interest.