We're skipping a lot of discussion to focus on the UK, which has arms measures that exceed (in some, but not all, cases) even the far-fetched hypothetical I threw out above. Shinzo Abe is not a black swan in the context of Japanese political history nor the history of political assassinations generally, but I digress.
To answer the point, there is no technical limitation keeping people in the UK from building, creating and shooting homemade or otherwise improvised guns that I am aware of.
What the UK does have is universal healthcare, a 3-4x lower incarceration rate and dramatically improved social safety services.
I think you can group the majority of shooters into three buckets -- ideologically driven (think white supremacists, Islamic terrorists, anarchists, etc), the mentally ill, and the criminally motivated (gang shootings mostly). The US has only amplifying factors for all three groups.
For idealgoues, there is no wider span of acceptable discourse than in the US. Commonly espoused views in the US legislative and executive branches are criminal offenses in a number of peer countries, e.g hate speech is still constitutionally protected speech in the US. The rhetoric is insane, accusations of nazism, faciscm from the left and similar accusations from the right, and generally a very high degree of polarization.
For the mentally ill, the support system in the US is abysmal, with cracks big enough to drive a truck through. There are multiple books written about the failures of America's mental health system, I will not belabor the point.
For the criminally motivated, gun crime is concentrated in young, mostly black men in decaying post-industrial cities in the midwest and (south)east. They have almost zero political capital, low social mobility and very little pubic support. Other countries certainly have their ghettos, but take a trip to Gary, IN or Jackson, MS. You would be hard pressed think you are in the richest, most powerful country in the world.
Fundamentally, the point still stands. There is not a feasible technical path to keep firearm technology out of a massive number of hands. The skills needed to produce a functional firearm have never been lower, and they will keep declining until almost zero. The only technical (preventative) measures run squarely into the bill of rights -- think a lowered bar for a warrant or infringements on the 1st amendment limiting the sharing of technical knowledge. Changing the culture -- around mental health, around poverty, and around power is very difficult, so we will see an attempted erosion of civil liberties, just like 9/11 was used to erode civil liberties with the introduction of the Patriot Act and similar legislation.