If guns are not allowed in England, can a bobby carry one? If not, what happens if a criminal gets his hands on one, what do the police do?
Specially-trained officers carry guns when required.
They’re called ARU’s (Armed Response Units), and as far as I know, they’re permanently driving around specific areas of major cities waiting for a call.
They’re also armed with more than a handgun - usually MP5’s.
About 1991 I was in the Strand in London when I saw a policeman loitering around a bus stop, his left hand trouser pocket was modified with suede leather - and he had an automatic pistol in it.
British police do not generally carry guns, but in certain areas in London or when undercover in places like Gatwick … I would not count on it.
The Metropolitan Police Specialist Firearm Command. They provide the rapid armed response (as mentioned above) and train firearms officers from other units (Flying Squad, Royal Protection, Diplomatic Protection and Special Branch and a few more).
I think you’ll find that armed police are present at most if not all airports
(Bolding mine)
I think you’ll find that guns are indeed allowed in the UK- but there are a lot of legal hoops to jump through to own them. Shotguns are, AFAIK, not as tightly controlled as rifles, and it’s almost (but not quite) impossible to get a licence for a functioning handgun.
Shotgun licences are pretty easy to get hold of AFAIK. Just about any farmer will have a shotgun, and game hunting is still widely practised. Handguns, as you say, are a different matter (since 1997).
There are cops with guns aplenty in Westminster where I work. On the way to my office I nearly always walk past two police officers toting large and mean looking guns hanging around their shoulders. I’ve kind of reached a point where I see it as a reassuring sight now, protection from the enemy and so forth, but I’m not a fan of guns and would prefer to live in a world where they weren’t necessary at all.
Thanks for fixing the title (changing the misspelled “Bobbs” to “bobbies”. It bugged me at first because I couldn’t figure it. Then my libidinous mind kept changing it to “Do boobs carry guns?” I hope not.
I was surprised to see pairs of Met officers with H&K submachine guns, walking along outside the Natural History Museum in Kensington a few weeks ago; I later realised that the French Embassy is pretty much right across the street (and there was a big lineup of what I assume were French nationals wanting to register for absentee ballots).
Anyway, anyone who’s seen “Hot Fuzz” knows that it’s in the country that everyone and their mum is carrying. 
Police officers acting in their course of their duties can possess or acquire firearms [not surprisingly]. They don’t get an ‘out’ from the laws off-duty, though, any more than any other citizen.
If the presence of an armed perp is confirmed (or suspected) they will contain the situation as best they can and summon an Armed Response Team. These will usually arrive far too late to shoot anybody, unless it’s a siege situation.
In recent decades the rather casual way firearms were handed out when the situation demanded to officers who had ‘been in the army’ or done a course some years before with a one-day refresher now and again, has been considerably tightened up and restricted to people with a permanent assignment to such duties [inasmuch as anything is permanent in the police as it is policy to shift people after a few years whether they want to go or not to prevent a ‘cosy little crew’ from developing]. Thus the number of individuals allowed to carry has fallen while their competence has [supposedly] gone up.
Going back into ancient history, pistols were quite frequently dished out to those who wanted one on night patrols in the '20s
I remember seeing MP5-toting cops at parliament and on my way to the Guards Museum. I believe we were walking past an important ministry (MOI or some such) when we were scrutinized by a gun-totin’ bobby guarding a heavily gated underground parking lot. But we were on a back street and were the only thing for the guy to even look at so what could we expect?