Old BA First is not worth it IMHO. Check the seat maps on the BA flight, if the economy seats go to 50+ then you have the new BA First on the 773. If not then you very likely will have old first.
Tell us more about the BOM-MAA flight. Airline, class of service, does it differ if you take the BA LHR-BOM leg?
Jet does operate a lounge in the Domestic terminal and it opens at about 2am.
BA New F is not route dedicated and aircraft can be substituted.