I actually think that a British Fallout would probably have a more post-war 60-80s vibe, given that those periods are much more relevant to modern day Britons than the Victorian/Edwardian era. If anything that time period that had the greater anxiety about imminent destruction, given both the Cold War and the decline of the Empire.
Remember one of the key differences between British and American pop-culture during this period is that each was deeply influenced by very different geopolitical climates. The US was entering onto the World Stage as a major power, and it seemed as though the next big conflict would break out between them and the USSR. In contrast, the British Empire was on its last legs, and Britain was resigning itself to taking a back seat. American fiction, and science-fiction in particular, tended to be optimistic and forward looking, whilst British sci-fi tended to be darker and more cynical. Another difference would be America's pioneer culture and the influence it had on Fallout. Fallout has a lot of cowboy imagery and tropes, as well as the element of trying to survive and thrive in the uncivilised wilderness. British fiction doesn't really have that element, and in any case it doesn't really fit with British geography.
Another thing that I would expect to influence a British Fallout would be the tradition of the
cosy catastrophe in British apocalyptic fiction. As such, in between the decay and ruins there would be almost absurd elements of normalcy.
Finally, I would expect a British Fallout to be even campier and more tongue in cheek and twee than OTL Fallout.
So here's what I imagine what a British-made Fallout would look like.
The West Country and South Wales are under the control of a warlord who styles themselves as King Arthur and who is advised by a mysterious figure claiming to be Merlin. It's later revealed that Merlin is an AI who thinks that humans cannot be trusted to govern themselves and is seeking to impose feudal order for the greater good. Arthur and his henchmen are either in on the plan or are robots or something.
In North Wales the people are fighting a guerilla war against Arthur, possibly using the old mines as vaults. Anglesea/Ynys Mons is controlled by neo-pagan Celtic revivalists, trying to bring back the Old Gods through human sacrifice and led by an Alistair Crowley type.
The centre of London is now a vast lake after being nuked, but the outskirts survived. The West fell under the control of a charismatic Thespian (voiced by either Judy Dench or Patrick Stewart) who has gone Richard III. The East fell under the control of an alliance of criminal gangs and remnants of the police force.
Briton is controlled by a gang of viscous Mods led by an
Alex DeLarge expy.
Essex and Northern Kent are under the control of National Front types.
Sussex, Surrey, and Southern Kent are under the control of French/EU peacekeepers trying to establish order. Their leader (who is short and has an outrageous accent) is trying to find any excuse to try and conquer the British Isles.
Manchester managed to survive the apocalypse largely unscathed and with much of its technology intact. As such they are currently trying to restart the industrial revolution. It's the most Dickensian feeling place, with just a hint of
Terry Gilliams's Brazil thrown in for good measure. Surrounded on all sides by warlords, and facing internal strife, there's a strong sense that they are the last beacon of civilisation about to be snuffed out.
Leeds is a workers republic under the command of totally not Arthur Scargill.
Liverpool is a wretched hive of scum and villainy. Also in this version it's become a hotspot for pirates who control the city. Among the pirate factions within, one of them is led by an
eccentric who styles himself as the King of the Pirates and used to be based on Cornwall until Arthur took over, and another one
has access to a submarine and dreams of launching an invasion of America.
York and the surrounding areas are under the control of a self-proclaimed King of the North, who has a Viking aesthetic and is voiced by Brian Blessed.
The Isle of Man has an Island of Dr. Moreau deal, inhabited by intelligent mutant animals, with the Moreau expy voiced by David Attenborogh. I could also see expys of various animals from British pop-culture, such as Aslan, Dangermouse, Bagpuss, Wind in the Willows, and Basil Brush/Fantastic Mr. Fox.
In Northern Ireland the troubles have been jacked up to eleven, with protestant and catholic warlords tearing into each other.
Scotland is divided between forces loyal to the British Monarchy based in
Balmoral, and Scottish Nationalists.
Other ideas:
Ruthless biker gang led by the
Chuckle Brothers.
Cannibal
Delia Smith.
Murder Mystery side-quest in a hotel whose manager is voiced by John Cleese, combining Agatha Christie with Fawlty Towers.
Rothamstead Research Centre is overrun by
Triffids.