The most likely reason for any increase is that, as has been frequently noted, people have not kept up medical appointments such as screenings and mammograms during the pandemic. There’s been a lot of catch up, in a short period. Vaccines sometimes have side effects - even the flu vaccine can cause side effects in a very small proportion of people - but the vetting procedures we have in this country means that the chances of a vaccine causing something like cancer are negligible. You’ll remember that the clotting problem with AstraZeneca was known, and publicised, very early.