We will be flying into and out of Terminal 5 at Heathrow. Now, my wife has taken fancy to a certain duty-free store located at Terminal 3. Assuming we factor in enough time before boarding our outbound flight from Terminal 5, are we allowed to enter Heathrow from Terminal 3, shop and then shuttle over to Terminal 5?
Short answer - no, not any more. (It used to be possible pre-9/11, but isn't allowed now)
To change between terminals airside at Heathrow, you need to go through Flight Connections. After the bit where they can issue boarding passes if you don't have one, is the part where they check your boarding pass. No boarding pass for a different terminal, no access to the airside shuttle bus.
Alternately, if you're allowed to enter the UK, you could enter the UK in T5, take the free train connection to T1/T2/T3 and walk the couple of minutes from there to T3. However, you'd then be stuck, as they would want to see a boarding pass before letting you into the security queue, and you won't have a T3 boarding pass, only a T5 one, so they won't let you in.
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...!