Nope. On October 23, 2077 every country that had nukes fired them at their respective enemies. Other conflicts just weren't explored in detail because they weren't relevant to the Fallout games to date. And we don't know if nukes were actually being used in the Resource Wars between Europe and the Middle East.
What? We do know... actually:
2053 December: Like an exclamation mark on the end of a very bad year, a terrorist nuclear weapon destroys Tel Aviv. - Vault Timeline
2054 January: Limited nuclear exchange in the Middle East raises fears throughout the world. - Vault Timeline
The fact that the European Commonwealth used "limited nuclear exchange in the Middle East" forced me to believe they fired nuclear weapons into the Middle East. The Middle Eastern powers, if it would even be possible to call them that, did not respond in kind with their own nuclear weapons except through terrorist attacks on Israel, possibly still under British control.
The FO1 and 2 intros are pretty clear in that all of the civlized world was nuked, and most of humanity was killed.
They aren't clear, at all, about how the civilized world went, actually. In the Fallout 1 intro it mentions "most" of the world being reduced to cinders, not all of it, which can presumably mean just about anything, it isn't clear on who fired nukes or what was nuked or if "cinders" even means nuclear fires. In my mind it refers to radiation after the Chinese and the Americans launched nukes at each other, which would have created worldwide fallout, this would have "burned" away the ecosystem and people (which is actually substantiated in the Fallout 2 intro "their spirits becoming part of the background radiation that blanketed the Earth"), while China and America would have almost nothing left standing from the bombs impact. After the brutality that I presume was used by all of the sides that participated in the Resources Wars I came to the conclusion that the bombs used most extensively were neutron bombs, (why would you destroy what you need with out-dated atomic bombs?) which would explain why there are still standing structures in places like DC.
I'm not saying atomic bombs weren't used at all, look at the crater in DC for instance (obviously, it is from a direct hit from an atomic bomb), the Chinese would have wanted to extinguish any places in the United States that would have posed a threat to any future invasion. They wanted to destroy urban centers (urban warfare isn't fun), military bases (a direct hit with a nuclear bomb can rip through the surface and into underground complexes), but leave in tact vital areas where they would need to get resources from, like mines and oil refineries, thus they would have used neutron bombs. Neutron bombs would have released much more radiation, as well, than would a normal nuclear bomb, thus it would have led to a much more severe worldwide fallout that falls much more in line with what the intros describe, (you know, humanity on the brink of extinction and such, it would also explain those who died in the sewers and the metro).
You're advocating a kind of 'global Vietnam' whereby the U.S. props up old colonies against communism? Plausible, given the U.S. we see in Fallout. However, they'd be mad to do it in the name of prolonging dying empires. The real life U.S. could care less about the French claims to the colony of Vietnam, they cared about, well, it's debatable, but I'll say - killing communists. This on a wide scale could indeed lead to what we see in the Fallout universe given the 'Red Menace' culture.
Might I state a small point of fact? The United States has never helped "prop-up" colonization unless we were doing the colonizing. Not only would it never (and never did) serve our interests, but it is against the sincerest American ideal of self-determination. The divergence was after WWII so self-determination would have been enshrined in American diplomatic policy. In the Vietnam War we fought on both sides of the conflict, helping the nationalist defeat the French colonizers and then fighting with the nationalists (those, at least, who we believed should be the true nationalists) to fight the Commies (who confusedly were the nationalists) in the name of democracy. In Vietnam, unlike in the rest of the world where America fought like in Korea, America confronted for the first time a place where the people actually wanted to be Communists. Communism wasn't forced on the Vietnamese, like in Eastern Europe, if anything, capitalism was. I see no reason why the United States, with a lot of its own problems, would even think about helping those in the European Commonwealth hold onto their empires, it would make no sense.
If there was a British Empire around the time of Fallout 3, I think that they would have been a HUGE target by other countries, even equal to how much of a target the USA was as Britain would still be a superpower.
There were four superpowers in Fallout lore, the United States, the Soviet Union, Communist China and the European Commonwealth. If the British stood as their own superpower I doubt they would have joined the European Commonwealth, in our timeline the British didn't want to join the early EU because they had their own commonwealth of nations, there really would be no incentive for them to fight alongside those it would feel would drain from their coffers.
In the real world it still was even, but not as much as the Soviet Union or USA as it was weakened greatly by WW2. But I doubt in the Fallout world it was an empire, it still had much control over India but by the 50s they were independant from them although still allies. The only land the UK had control over was the British Isles and a few islands dotted around like the Falklands. But still, the nuclear war happened around 2077, well over one hundred years for the UK to recover from its weakness and Europe to recover from WW2. The Resource Wars did weaken Europe to fight each other but the UK would have had the advantage of being an Island, giving it a lot of strength and defence.
In that case, why would the UK want the headache of an Empire again, they could seek those same resources in Europe and gain those resources much quicker as they would actually be developed.
The Royal Navy was still the premier naval force and any invading countries would have been easily destroyed.
After World War II, most of Her Majesty's Royal Navy was disbanded so the British could pay their expensive war debt, the colonies, that would have been a burden at this point financially before resources were scarce way after WWII, were given up in order to pay for that debt and the British occupation of Ireland was ended in order to disband a huge chunk of the army. In a capitalist world economics drives decision making, I see no reason why that would have changed in the Fallout universe.
The only threat UK would have faced would have been within and it would have been plausible for it to split once again into Scotland, Wales, England and Northern Ireland which would have great instability and an easy target for invasion should this have happened.
It is possible that the United Kingdom split apart. It also possible, that like in the first Great War and World War II, the country stuck together to fight its enemies. I doubt, though, the British would have been able to hold onto Northern Ireland, with how violent the IRA is. It would have made no sense for the British to obstruct Irish unification, wasting valuable resources for very little pay-off.
If there was a nuclear Holocaust across the world, then Britain would still be targeted regardless of its strength as it was a key player in international affairs and would have many enemys eager to nuke it. Britain would have been bombed to bits much like other major European countries like France, Germany. Spain, Italy etc . Europe was the heart of the world and a major political area and America was and is still largely economically dependant on Europe. If the EU was formed and didn't fall apart then it would have been a global superpower to rival America and China, although in the Fallout world this wasn't the case. China would have nuked Europe to have effects on the USA
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Heart of the world? Maybe before 1900s, but by 2077 Europe was in full decline in being a player in international affairs, the European Commonwealth dissipating was the last gleans of glory for old Europe. A Europe after the Resource Wars, the brutality involved, would have made a very unusual target to attack. I like to think people are a bit more logical than they are given credit for and a nation would only use nuclear arms once it knew there was no other way to stop an advancing enemy. Once the US was on the verge of occupying all of China, the Chinese launched their missiles at the United States and vice verses, ending the world turning it into an irradiated mess.
The premier naval force post WWII is the United States Navy. There's no contest. In the Fallout timeline we've no idea how strong China's navy is, but my guess is it's at least as powerful as the U.S. Navy to be able to pull off the invasion of Alaska.
The Chinese, if they launched a surprise invasion, with their far better submarine fleet protecting the ways of supply, could have easily pulled it off establishing a beachhead and securing Alaska before the army would have even been able to respond. You know, Japan had taken Kiska island on the tip of Alaska during World War II, we didn't figure that out until late 42' after a month or two occupation. Alaska is very remote, we Americans sometimes forget it even existed, I see little reason for American attitudes toward Alaska to have changed that much after World War II. The only thing we did to fortify our position in Alaska due to the Japanese invasion was to build a highway through it, that doesn't mean we put sophisticated radar stations all down the coast.