I agree that vaccine mandates are a mistake - they will only harden the opposition.
I would start charging the unvaccinated extra for their health insurance, or make them pay some nominal sum to the NHS if they end up hospitalized - like 100 quid a night. There's probably a pretty strong case to stop paying unemployment, long term disability costs etc to folks who end up off sick for weeks or months and are unvaccinated.
Many US employers who self-insure for healthcare are charging unvaccinated folks (without a genuine medical waiver) extra. Seems fair enough to me, since my three shots cost a total of about $150, but a hospital visit will cost anywhere from $5000 to $500,000 depending on how long you're in.
Personal responsibility - it's what the right wing is constantly preaching - time they walked the walk.