I had a poor experience with the personal shopper, would this still work as I want to go to T3 but both my BA flights are T5 and shop at my own pace:
"There is a workaround though if you want to do a (significant) amount of shopping. As described in
this FlyerTalk thread at the end, Heathrow provides a
Personal Shopper service. If you contact them in advance about it, they can apparently arrange for the item to be brought to another terminal so you can look at it and purchase. Apparently, if you have a lot of time, and it's quiet, they might be able to arrange a transfer, but that part isn't guaranteed.
That looks to be your only way. Well, other than booking a fully refundable (check the T&C closely!) ticket out of T3, follow flight connections to T3 for that, shop, go online and cancel the ticket, ask customer services in T3 to "land" you, follow flight connections back to T5 on your onward boarding pass, then collapse in a heap on your return...!"