@JingsMahBucket,
I'll put it in simple terms, as the OP has suggested.
So we roll out a vaccination program quickly. We are called into our doctors, health centres, walk ins and schools, as well as other centres, mobile groups, so that people without local GPs, non British people, homeless, travellers and everyone who is in the UK, because not everyone lives exactly like the OP does, in a permanent brick built house having been brought up in the same, here in the UK.
So we have an army of people who since the crisis started have been pushed to the limit.
So on top of giving out the vaccines, they have to check if the person has had the MMR and then ask them why. Is the truth going to be told do you think, or will they lie to get the vaccine?
Do we make people get the MMR first and so they come back in two months, but we protect their children meanwhile?
Do we do that with visiting/short stay/refugee non British people, if we don't is it fair to deny our own citizens the same?
If the homeless people aren't trackable and they don't know, again do we ignore that some people haven't had the MMR?
Or do we just do the sensible thing that will get rid of this virus as quickly as possible and just give everyone the vaccine? That includes no questions asked so those in the country illegally can get the vaccine, so this never returns.
The OP seems completely unaware of the different demographics that make up the UK. Then you understand the complications of putting conditions on something so vital.