Personally I find transit through London is rarely stressful, and I do it quite a lot. The thing that would stress me is that 4 hours is too long! 2 hours is about perfect in my experience. Even if you do miss the connection - somewhat unlikely - then provided it's on one ticket you will simply be rebooked on to the next service.

Now make sure you book a pure BA connection (or AA codeshare using BA metal) to Bombay/Mumbai, this will keep the transfer to T5 alone. If you end up in on AA metal going to LHR T3 then you do have a slightly more awkward transfer process there from T3 to T5. (AA don't fly from PHL to LHR). Even then 4 hours is plenty.

You will go through security when you get to T5 on either route. Depending on your nationality you won't normally need a visa, and there's no official passport check on this transfer (apart from BA checking your id matches your ticket name), so long as you stay airside. Your bags should go all the way from PHL to India, even if on 2 tickets, provided the flights are oneworld.