I really don't understand where you get your ideas from about doctors, lawyers and accountants.
If you are willing to spend the time studying to attain the necessary skills to join one of those professions, then they will welcome you with open arms.
Do you expect them to accept Joe Bloggs, Bricklayer, into the Medical profession?
I know dozens of accountants, and whilst, in general, they are the most boring people I know, I have no idea why you think they wouldn't allow you into their professional body if you were suitably trained.
Solicitors have changed slightly in the last couple of years. Now, non-solicitors can finally own companies that do law as a trade. Barristers on the other hand haven't changed a bit, they're still stuck in the 80s, the 1880s that is: unless you're one of a trivially tiny minority, you're not getting a barrister's pupillage unless your face fits and you're of the right social background for them, the city ones are the worst for this by a long way in their snobbery.
Doctors, lawyers and accountants may not be 100% fits for the closed shop definition but their self-protection beats the unions of the 80s hands down. Just try to get a lawyer's practice certificate removed, or a doctor's, for proven gross negligence and you'll find they'll protect their own like in all bar the most highly publicised of political cases with a serious crown court conviction forcing their hand. It's only recently that lawyers were 100% in charge of their own discipline as a profession, now it's only 50% but they're still nearly immune to challenge. Try suing one, good luck in winning anything important.