UK has about 12 million aged over 65, a million or more frontline NHS staff, and a lot of care home staff. No idea how many 'vulnerable' to add on. I'd guess in all it could be as much as 20% to 25% of the population, say about 17.5 million people, needing 35 million doses of vaccine to be given.
The Oxford/AZ vaccine, despite being easier to produce and transport, looks like it might be in fairly short supply, at least in the UK.
This article comments on some of the issues it's had ramping up. Summary is that of the UK's 100 million dose order, about 40 million is/was expected to arrive by the end of March. I haven't looked up how many of the other vaccines we are expected to have but from what I recall it's a smaller number.
So, I dunno... end of February at the absolute earliest for us (UK)???