Will the COVID-19 vaccine be compulsory in the UK?

The vaccine won't be compulsory here in the UK.

Some, stupider, people seemed very concerned about this even before there was any sign of a vaccine. I've never heard any suggestion that vaccination would be compulsory nor can I think of how it could be made compulsory.

I would hope it's likely though that people who refused to vaccinated would be limited in what they could do and where they could go.

Edited after reading your update. Any reasonable person will be against compulsory vaccination. However it's perfectly reasonable that those who refuse to be vaccinated should be restricted in what they can do and where they can go. To paraphrase your phrase, if the unvaccinated are no threat then why do we need a vaccine?

This should be obvious: if unvaccinated people are free to do whatever they want and go wherever they want then it is likely that anyone invected will infect other non-vaccinated people. A restaurant or pub, for example, that allows one unvaccinated person in must allow all unvaccinated people in. The health service will be spending time and resources caring for them. Whilst there are unvaccinated people Covid 19 will be able to survive, spread and evolve. It would seem almost certain that a strain would evolve that was immune to any vaccine we have. It would also be likely that, over time, the number of people deciding “not to bother being vaccinated” (and it's likely NOT to be a case of one single injection that lasts forever) would increase. I don't have the figures but I know that some diseases that were virtually eradicated are on the rise again because of misinformed parent deciding not to have their children vaccinated.

Your argument would apply to people being allowed to travel in a car without having to wear a seatbelt or ride a motorcycle without wearing a crash helmet. No one is forced to do either of those things but if they don't, there are restrictions on what they can do - they can't travel in a car or ride a motorcycle.
 

Unfortunately they cannot make it compulsory, but they can make it compulsory for anyone who wants to fly or go to school, or go to places where people are a danger to others. 
Other diseases like measles could have been wiped out completely if everyone got vaccinated, but there are always some who have no commons sense or are afraid of needles. 
The covid vaccine is safe. Covid is not. I prefer the vaccine.