No, it shouldn't be obligatory, that's a very dangerous route to go down in terms of precedent setting for (future) government(s). But educational campaigns, clear information distributed in multiple languages are vital etc and need to be carefully targetted. The messenger is also as important as the message, so as
@LutonSpurs says, faith leaders and community leaders have a pivotal role to play.
I think it's likely for this year and into next year that there will be certain activities which will favour those who have had the vaccine, overseas travel being the obvious one, where evidence of vaccination will become a substitute for quarantine and pre-travel covid tests. But that still leaves people with a choice for people to make.
And to your second point, the NHS, and care homes in particular, would struggle to operate without the workforce that comes from many of the groups currently distrusting of the vaccine. (I agree that everyone working with vulnerable people
should have the vaccine, but it mustn't be forced on them).