4 hours is plenty of time to make the connection and your bags should be checked all the way through to BOM. I don't think AA flies to LHR from PHL so that should be a BA flight so no terminal transfer. You just have to reclear security.

I would consider the non stop via EWR if you can lug your bags there and prefer a non-stop. Personally I prefer to stop somewhere and have 2 long flights instead of having 1 really long flight and LHR is great for that.