I don't think I'd go out of my way to upset a doorman. Although their trade is regulated (in the UK) they are unlikely to come from a legally trained background and, putting it politely, they may act first and think later.
That said, I have absolutely no concern about acting within my rights in the presence of UK Police and we should actually be encouraged to do so. As Aksel's anecdote shows, Police are there for our protection as well as to uphold the law if we're breaking it.
The only issue, photographically, I've ever had with Police relates to PCSO (Police Community Support Officers). They dress "like" the Police but have considerably less powers of arrest. I suspect, being a relatively new outfit, they are probably the current government's attempt to be able to claim that they've increased the number of Police on the street.
A couple of them have "interrupted" me whist taking photos in London and even attempted to get me to show them what I'd photographed (difficult without removing the film and having it processed) but, as they're not entitled to do so, they were told to sod off I declined. One even tried to insist that I hand over the camera or the film.
I definitely wouldn't have any problem with photographing the Police whilst they were performing their duties (without interfering) as, if they're acting within the law, what do they have to hide?
