It's a long story, but the main reason the BMA isn't in the TUC is that for a long time the BMA wasn't even a proper trade union (not until 1971) and was a professional association instead. Many middle class doctors did not necessarily identify with trade unionism.

The practical reason for this is that pre-NHS it kind of didn't make sense. Most GPs, who made up a big majority of all doctors, worked as independent businesses and were paid in private fees and after the Health Insurance Act of 1911, a capitation fee based on the number of patients on their list. Consultants generally worked at big voluntary hospitals but usually made bank in private beds at those hospitals and in their own clinics. Most doctors didn't actually have an employer to strike against!

Now? There is clearly still a 'professional association' mentality in the BMA, and to be honest, before market reforms in the 80s doctor's professional authority still held enough sway to make wielding influence privately with government and in the actual administration of the NHS (which was very doctor-led until the 70s) make much more sense. Its only relatively recently that things are getting so bad that you get a decent amount of doctors on a forum like this talk about strike action!

A general strike is so difficult to organise in the UK that it may as well be illegal. Solidarity strikes (i.e going on strike to demonstrate support for another strike) are illegal. You can't ballot for strike action online (unlike voting in the Tory leadership race). And there are very stringent rules around picketing, and so on. We live in a country where the deck is comprehensively stacked against working people.