With regards to the vaccine rollout; to put things into some perspective, approximately 3.5 million people have now received a first dose in the UK, and 324,000 people were vaccinated yesterday alone.
Even if this pace remains roughly constant, I calculate we could have vaccinated the entire UK adult population with at least one dose (best number I can get for this is approx. 52 million) by 16th June 2021. (There are approximately 48.5 million UK adults left to go, and 48.5million/324,000 = 150 days to the nearest day, and 150 days from today is 16th June 2021)
Based on the fact that the vaccine rollout will probably only accelerate from here due to things like 24/7 vaccine rollout coming online and new vaccination centres opening, I think offering the whole adult population at very least a first dose by June is very much attainable!
I think the September estimate might be talking about second doses/complete vaccination, as 12 weeks from 16th June does take us into September.
Based on this calculation, I should point out that the government also looks set to achieve its target of having all of the first tier of vulnerable people vaccinated by the end of February, as I calculate that this could have happened by 22nd February 2021 if the current pace continues.