Chennai (India) to Raleigh-Durham (USA) baggage booking

British Airways does not do a flight from London to Raleigh, I believe this is American Airlines, code shared with British Airways. Yatra, which is a popular travel website in India shows that the exact same trip as mentioned in the question can be made on a single reservation with Air India/British Airways with the flight operated by American Airlines.

Thus, I believe you won't be required to re-check in your baggage at Mumbai (also because Chennai is an international airport) and you should be able to directly collect it at Raleigh.

If you have booked your trip as one reservation then you do not have to worry about re-checkin. Also, British airways has an alliance with American Airlines. So I am very sure you do not have to re-checkin your bag in London.

If if you have booked the ticket through a travel agent, sometimes they book the tickets separately and then give you the final itinerary. What I mean by this is, if the agent booked your ticket from Chennai to Mumbai separately, and them made one reservation for Mumbai to London to Raleigh.... then you will have to rechekin at Mumbai. But if the agent makes one reservation for the whole journey starting from Chennai to Raleigh, then you do not have to recheckin at any point in between. I know this for sure because I travel between Chennai and Raleigh often. Good Luck With Your Trip!