This graph with data from gov.uk up to 25 September shows very clearly that the big rise in cases is in under 14 year olds, who haven't been vaccinated. The increase became steeper when schools in Scotland and some in England returned after the summer break. 15-19 year olds, many of whom have had at least one dose of the vaccine, show a steep decrease. Maybe it's a coincidence, but I doubt it.
I hope that answers the OP. Take under 14 year olds out of the equation and the number of new cases would be almost halved. Vaccinate them and the country really would be in a situation where Covid was a relatively rare condition and maybe, with a couple of precautions, the country really could get back to normal.