EU citizen here.
The EU has never tried to 'ban' use of the AZ vaccine. Moreover, the EU is not a state, and the country that banned exports of AZ vaccines (Italy) was not the same as the countries that started the (preliminary, now over) suspension of the AZ vaccine (Germany, among others).
There is nothing the UK has done about vaccination that it could not also have done while in the EU. Yes, there would probably have been pressure to distribute some of the doses now available to the UK to smaller EU countries - which in the big scheme of things would not be a bad thing - but the NHS was never under EU control.
The one constant, pre- and post-Brexit, in the UK is the piss-poor quality of debate about the EU and the rampant ignorance that permeates it. I should know, I lived in the UK for 15 years; left late 2016.
Imagine for a moment that the situation were reversed, with the UK short of vaccines and the EU not. Compared to the screaming and shouting Brexiters would be doing then, the EU is fairly low-key.
Here in Belgium, infections are up but deaths are still down. Vaccines may be few, but they've been targeted at the right people. Of course I wish there were more vaccines and less vaccine hesitancy here. But the way UK retainers are using the vaccine rollout problems in the EU to reconcile themselves to Brexit is, frankly, a bit pathetic.