corporate-wage-slave

It's not absolutely impossible. You would pass through Flight Connections from T3 to T3. You cannot do this once you have gone through the UK Border, nor can you re-enter T3, therefore you can't collect baggage either. You would probably be challenged by the HAL staff since there is a boarding pass check at the security scan, however it is legitimate from HAL's perspective to enter T3 to use the lounges. In this respect the arrangement is intended for the paid-for lounges and dayrooms, rather than using the oneworld lounges. It's possible that the oneworld lounges agents would then deny you admission if they thought you were in the wrong terminal, but equally they may not know / care.

Whether you did or did not get access you can then go either airside to T5 or landside - there is an exit to the UK Border by the AA lounge which is signposted. If you go airside - which is easier and quicker - should you have e-gate access it may well be faster to use the First Wing rather than Flight Connections. However you do it, you do need to clear conformance at T5 at least 35 minutes before the Oslo departure, and you should allow 30 minutes for the T3 to T5 transfer, so you should leave T3 over an hour before departure (MCT is 90 minutes, by way of a guide). It should take less time.

Personally I think this is all a bit unnecessary, given the lounge provision in T5, but it's not really for me to judge. There would be two security scans doing this, rather than just one.