Mutations have been occurring constantly; we didn't do much genomic analysis before, but we do now. Many countries don't do this, even now.
People forget that Wuhan battled a less transmissible form; China's incredibly harsh lockdown helped but it wasn't the only factor; Europe had the much more transmissible G614 variant ( see
https://www.railforums.co.uk/threads/what-sage-really-think.212718/#post-4925455 ) in what is referred to as a 'D614G' mutation.
People who say we should be more like China conveniently forget that this mutation had occurred so that Europe fought a much more transmissible variant, and that a like-for-like comparison isn't possible (or maybe they are just ill-informed in which case they ought to keep their traps shut and leave it to be people who do have a clue!)
We are now seeing 'convergent evolution evolution' of mutations such as 'E484K' (The glutamate (E) to Lysine (K) substitution at position 484 (E484K) in the receptor binding domain (RBD) of the spike protein); K484 variants have occurred independently in different locations.
If a particular change is evolutionary advantageous for the virus, it will happen, and it won't just happen once.
The idea that there are an unlimited number of such mutations, and that they will likely occur somewhere else not within the UK, and that shutting the borders will keep them out, is utterly false and is peddled by people who lack understanding of how this virus is evolving. There are loads of such people spouting absolute nonsense right now, using this nonsense as justification to keep restrictions in place and falsely claim that vaccines are ineffective.
The reality is that E484 has occurred independently in Britain, South Africa, and Brazil. And potentially other places that don't carry out genomic testing.
Such changes do not affect T-cell immunity; they can impair the antibody response but they do not render vaccines ineffective. The vaccines can, and will, be tweaked to increase efficacy with these variants.