No, Brexit didn’t change the UK’s ability to authorise the Covid-19 vaccines earlier than the rest of the EU.
Boris Johnson was asked on the BBC's Andrew Marr show how "ordinary voters" in a place such as Leigh in Greater Manchester (which voted Conservative in the 2019 election) would benefit from Brexit.
Is the EU ‘pinching our fish’? - FullFact
21/06/2016
"[The EU is] Pinching our fish" - Boris Johnson quoted in The Telegraph, 16 June 2016