Probably. Many studies suggest that vaccinated people are less likely to spread the disease than unvaccinated. But the point about covid is that the virus appears to be at its most infectious in the very early days of the disease, so it's before the immune syste has really got going in getting rid of it.
Suppose the virus is at its most infectious in the first 5 days.
A vaccinated person may breathe in the virus on day 0, the virus starts multiplying but the immune system starts to get a grip on day 3, by day 6 the virus has gone before you ever knew it was there. An unvaccinated person has the same trajectory till day 3 but then the virus keeps on growing and by day 6 they feel pretty ill.
The point is for at least 3, perhaps 4 of the 5-days infectious period, there is little difference in transmission risk of the vaccinated v unvaccinated because the infectious period comes so early.
(This isn't an exact position re. coronavirus, just an illustration of how it might be that vaccination makes little difference to transmission. All day numbers are guesses based on what I have heard, not based on scientific study.)
Point I am trying to make is that NHS and care staff not being fully vaccinated increases the risk of onward transmission, although by how much is debatable. If people are arguing it shouldn't be mandatory for them from a freedom of choice perspective then that is one thing, but it isn't without downside if they don't.