Never really had an issue (12x this year), a couple of times the AA out of USA was late getting in, they will provide you with those orange ticket jackets, which helps get you thru,
T-3 to T-5, not bad, takes about 10-15 minutes, if you get a bus right away,
If you traveling thru LHR on Sunday, the transit time from T-3 to T-5 is even less, no lines at security.
I always do the AA to BA (going to BOM) transfer is smooth, just make sure, if you are checking bags, to have BA make sure your bags are scanned for the BOM flight
(twice this year, my bags, did not make the transfer in time)