There is a permanently armed branch of the UK mainland police - the
Civil Nuclear Constabulary.
Whilst there are a decent number of licensed FAC holders in the UK, the process of acquiring a licence or getting a renewal (every 5 years) is onerous (amongst the most stringent in the world) and requires quite serious vetting, including a doctor's certificate - if you're an FAC holder, that's on your medical record and doctors are required to notify the police of changes in mental health etc.
FAC licencing in the UK
The UK FAC also specifies the number and types of weapons and the calibres and quantity of ammunition you are allowed to hold at any time. Serial numbers of all guns and (where available) moderators (see below) are recorded. Also, how and where the guns and ammunition are stored is inspected and required to be a certain standard of security. It's a serious criminal offence to breach any regulations around an FAC. Moderators (silencers in the USA) are also regulated and licensed and individually mentioned on your FAC - you have to separately justify the need for a moderator.
As far as I'm aware, the last time a civilian FAC holder committed a crime involving a firearm in the UK was over 10 years ago (June 2010, Cumbria). The problem, if there is one here, is not with licensed firearms, but with illegally held and fake-but-authentic-looking weapons.