As usual, it depends.
If the AstraZeneca vaccine manages to do a more convincing job with the data it's got and with whatever it can assemble over the next few weeks - it could get its emergency use approval and start rolling out before the end of January. They think they can ramp up production fast so we might see all the highest risk (over 70s, healthcare workers and the clinically extremely vulnerable) done by early Q2. If we're relying on Pfizer/Moderna it takes the whole of Q1/2 to get there, and that assumes Pfizer/Moderrna don't hit further production issues. Better than nothing, and a massive reduction in the death rates, but still not great.
The other possible gap fillers are Novavax and J&J - if their current Phase 3 trials show the right efficacy/safety. Both of those may be ready for emergency approval late Q1. In theory at least both of those could ramp up production quickly. If those are available then Q2 could see a real increase in availability - probably enough to cover all the NHS priority groups including the over 50s. If that happens then by summer Europe will be a much more normal place - not business as usual perhaps but close.
In other words - fingers crossed for AstraZeneca (revised report!), J&J and/or Novavax.
The BBC has done a nice article at
https://www.bbc.co.uk/news/health-55274833 that includes things like the approximate number of people in each priority group.