mikeyfly is about right to be honest! The near term options are Thai direct (weekly from Heathrow) or Singapore Airlines.

There are due to be some "corridor" flights set up BKK-Phuket to allow transit (currently not allowed in Bangkok, you have to arrive to Phuket directly from abroad). If these were operated by Bangkok Airways then likely they'll carry the BA code and BA will sell cash tickets with the connection. But the BA flight leaves Bangkok to London quite early so depends if you can connect or not (currently there aren't any scheduled flights to BKK that would connect to it - though once you've been on Phuket for 14 nights you'd be allowed to take any flight to Bangkok and do an overnight/couple of nights there before taking the flight home)

The other usual "BA" way to Phuket is via Doha on Qatar (BA codeshare) but that relies on Doha coming off of the red list. If you're looking for winter you'd think this were quite likely but I wouldn't be booking yet.

Via Singapore needs Jetstar to be operating and BA to be allowed to transit. The Jetstar flights carried mainly local traffic and ex-Aus Qantas/Emirates connections, neither of which will be running for a good while yet so doubt that'll be a realistic option well into next year if not beyond.