In England, when private gun ownership was legal, contables used to carry batons. Now, private gun ownership is largely illegal, and police carry guns.
In Australia and NZ, it's important to note that you must register each firearm and you need a reason. That reason can include pest control or sport, but it cannot be self defense.
I think one of the biggest reasons firearms are not registered in the US is because the gun industry and the NRA lobby against it to keep guns flowing into South and Central America. I wrote about this a while back:
http://fightthefuture.org/article/america-and-the-mexican-dr...
Norway similarly makes it relatively easy to own a rifle or shotgun for hunting. But it is that much harder to get a pistol.
Best i recall, you need to have been a member of a target shooting club for some time before even being considered. And even then you are only permitted to transport it to and from the range, not general carry.
The rules are similar to the ones you describe for Norway.
There's also the "Armed Offenders Squad", maybe analogous to a SWAT team, who are armed. Don't know too much about them, you just hear them mentioned on the news when there's a gunman who has shot someone.
(Is this a language difference in "constable" vs. "police" vs. "cop" etc.?)
But keep in mind in the UK all weapons are prohibited. Walking down the street waving a baseball bat in a threatening manner is a criminal offence in the UK. Any object that could reasonably be interpreted as a weapon needs a lawful excuse to carry it in public, and a bunch of objects that violent people tend to fixate on are just outright banned, there aren't any excuses permitted for those, you just can't have them in public at all.
Thus having just the _same_ gear as the cop: pepper spray, an extensible baton and some hand cuffs, would get you in trouble here, never mind any sort of firearm.
"Constable" is the legal term used in UK law to refer to police officers and give them special powers or immunities. A Detective Chief Inspector is still a Constable by this meaning, so is a Special Constable (volunteer spare time cop) but a Police Community Support Officer (cheaper fake cops with less training) is not a constable. Outside of statute law, ordinary people don't call them constables.
British pop culture says the equivalent of a "get out of jail free" card would be membership of a Masonic Lodge. Police officers (in particular senior ranks) are supposed to be particularly likely to join the Masons, an esoteric order with a big emphasis on helping other members.
I was surprised that there were only 10 incidents where police shot their firearms in year ended March 2017 (although multiple officers might use their guns in a single incident). I guess some of the use of firearms is deterrence - a lot of police wandering round waving submachine guns at airports and stations when the terrorism threat is high.
https://www.gov.uk/government/publications/police-use-of-fir...
BTW, in recent years there have been a "temporary" general arming of the Norwegian police that have lead to a bunch of incidents (most of them accidental discharges as the issues handgun do not have the typical safety, as they were not intended to be carried on the hip for extended periods) including one where a officer fired a warning shot into the air...