Given that vaccination takes 2 shots, given 6-8 weeks apart typically, you’re already out of time.
Exception is the Janssen/Johnson & Johnson vaccine but that’s not been widely approved.
And then there’s the problem of finding a country that will a) let you in without being vaccinated and b) offers their vaccines to foreigners who aren’t residents. Very slim chance again.
And all that assuming you had the funds to pay for a) the travel to and from that country, b) the probable mandatory quarantine there upon arrival, c) the vaccine itself.
And after that you’d still be required to be in mandatory quarantine most likely upon your return home.
So no, it’s highly unlikely you’ll be able to get your vaccine somewhere else. All you can hope for is getting yourself on a priority list with the NHS somehow (and really, that requires medical conditions you don’t want at your age).