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Not officially. My father-not-in-law was determined to get Pfizer last year though so went around Walthamstow asking who had it until he got it, but he'd been living in the Philippines and had also had the Chinese vaccines, so was theoretically unvaccinated. He's also a boomer so they were desperate to jab him.
You could play the same card presumably, but there's only Pfizer, Moderna and Novavax available in the UK at the moment, and I don't know anyone who's had the latter. Not sure why you want to choose but the efficacy of Pfizer and Moderna are very similar and both mRNA vaccines.
If you're worried about Moderna side effects I felt fine after 24hrs and was never that bad.
I've had three doses of one of the Chinese vaccines but would like to get a western vaccine once I'm in the UK - can you choose which one you want?