Though the above is correct about inter-lining, because you are at HKG, using a Bangkok Airways flight, I think there is a very good chance that BA will interline in this situation, bearing in mind you are in First. A friend of mine did this precise connection quite recently and it was interlined, however my friend did ask in the Qantas / BA First lounge to ensure this had been done. Also my one and only flight with checked luggage in the last 5 years involved bags going on 3 separate PNRs from upstate Burma to Newcastle on 4 sectors with Thai to BA at BKK.

So in your shoes I would ask Bangkok to check through to LHR, go through the transfer passengers section, go to the Qantas First lounge, then check. If all is well you should have plenty of time to relax. If not, though it would be a nuisance (HKG separates arriving and departing passengers), you would still have plenty of time to go out and come back in.

Having said that, the Qantas First and Wing lounges are both by security on the upper departure level so if you do find yourself leaving airside to recheck, you are geographically not doing much of an extra walk compared to a flight transfer.