In terms of the vaccine roll out, I'd argue no, brexit or being part of the EU wouldn't have changed anything. Vaccine rollout was the UK's chosen strategy from the off (despite not saying it outright and incorrectly mentioning herd immunity then denying they said it). Each country chose how they wanted to manage this crisis and did so according to their own laws and principles, as has always been the case.

Being a member of the EU wouldn't have changed that as the UK (and all other countries) are able to make decisions as they see fit. In terms of brexit, for individual people the govt has made sure that the general public don't feel the true impact (queues at airports etc) as those small annoying things are the only decisions that most people will ever feel of the impact of brexit directly. So for most people they won't ever notice a material difference, at least not immediately.

The problem will be for businesses, and the medium to long term, especially with the supply of food and products and the standards which these are set at, which will mostly be lower. The general public have no control over these so are at the mercy of the govt. So if the govt buy chlorinated chicken from the states, the consumer can't stop that deal and will struggle not to buy it in the shops. Slowly but surely they'll just accept it through lack of affordable alternatives. But that's a 10 year plus plan.