Assuming the other suggestions don't work out: Four hours would be a good answer. Potentially you may be able to do it in under 90 minutes if the Gods smile on you, but you would need to do this landside. So it's T5 arrival transit (probably) to UK Border for passport control - collect bags - Heathrow Express train to T3 - recheck bags - security, gate for departure. Doing it all on BA would save a lot of hassle. If you allowed 4 hours and had good travel insurance then that would be valid as an approach and you'd probably have 2 hours airside in T3 (which has the usual shopping mall set-up).