// But the Astra-Zeneca vaccine was the first to be administered in the UK, while the EU were still arguing about it, and before Pfizer was rolled out. If I remember correctly, Pfizer was the preferred option of the EU, and they were still arguing about the terms of its distribution when UK started using Astra- Zeneca. //
Some of this may be true, but it was a political choice for EU members to coordinate, rather than a necessity. The UK could have taken the same path it did independent of Brexit -- perhaps pointing to, for example, its non-membership of Schengen, its status as an island and so not part of mainland Europe, etc.
It's true, in other words, that the UK was faster than the EU to roll out a vaccine -- and I'm glad that they did -- but that was not linked in itself to Brexit.