This is if you have a bag to check-in.
On arrival at T2 you need to head over through immigration, then pick up your bags, then exit the building and descend 2 levels to the underground part of the Central Terminal Area. Here you have to choices:
- If you have a contactless credit/debit card or Oyster, you can take the Underground to T2. The journey is free of charge, but you need a card to open up the barriers. A child can go through with you (up to a certain age, then he/she needs to get his/her own card). Travel time T2-T5 is max 10min, though you might need to wait a little bit. Check TfL.gov.uk for future engineering works, they're normally on weekends and sometimes they can mean that there's no service in the LHR area.
- Take the Heathrow Express. Once again free of charge. This isn't an option I'd use because the HEX is less frequent, the station is further away from T2 and it takes you pretty much where the Underground takes you. Anyway, if you really want to use the HEX you need to walk further along on the tunnel connecting T2 and T3, make a right when indicated, either take a (free) ticket or (I think, but not sure) use a card, then elevators down. Again, I wouldn't use it unless the Tube isn't running.
Once in T5, head upstairs to departures. Both for the Tube and HEX, do not use the escalators, use the elevators (aka lifts). Lifts don't need to be called, or buttons need to be pressed, they just go up and down automatically. They do get crowded, so if there's more choose the one furthest away from the masses, that's my trick.
In T5 you'll need to re-check your bag, get your boarding pass, clear security and go to gate.
Time to do this whole thing: that depends wildly. T2 immigration gets
very busy when all the US flights arrive. Took me often more than one hour for the e-Gates. You said you have a child in tow; if he/she is younger than 12 then you're out of luck and can't use the e-Gates. If your passport isn't British/American/Canadian/European/a few other nations the wait can be L-O-N-G. Basically, if I were you I'd give myself 3-4 hours between one flight arriving and the other departing, just to be on the safe side. Better being bored than harried.
If you don't have a bag to check in.
This would be my suggestion. Don't check in a bag. In this case, assuming you already have your BA boarding pass (downloaded through the app), then what you need to do is, on arrival in T2, to head over to connections (purple signs) and hop on a bus to Terminal 5. Journey's about 40min. Once there, don't go to immigration but head over to connections again. There you'll go through all the necessary step, including document checks, a security scan and then you'll be in the departures lounge. This could take as little as 1h30.