At the risk of repeating what Dan said, my Internet has been going all goose-poop so I wasn't able to post this last night:
From the focus of an empire upon which the sun never sets to an island upon which hope never dawns? Post 1900? Whew... Well, here's my stab at it.
We should probably start as soon onto the new century as possible. I'd say the best place to start would be troubles in Ireland. If you can accept some cocked up version of Home Rule passing pre-1900 but holding together until after the new century, you can probably start with a freshly topped off cup of resentments just waiting to be poured out. Home rule was again being...uh, discussed, shall we say?...as Europe lurched into WWI, only to disappear in unity to the war effort. Instead, Irish Nationalists use it as an opportunity for trying to independence. They don't succeed but it requires a lot of troops taking a lot of casualties in Ireland at the same time as on the continent. Great financial and manpower stains throughout the whole 1914-1918 period.
Use that to set up a more fractured political climate. Say, 1/3 of the electorate that believes Britain should focus on Ireland, even if it means drawing away from efforts at the Western Front, 1/3 that believes the Western Front is paramount and must take precedence over Ireland, and 1/3 that believes that both must be pursued with full vigor. This gives any Prime Minister the dilemma of having an electorate that will both agree with any policy, in part, by a comfortable majority while still disagreeing with it, in the whole, by the same margin.
OK, we work it so that the "all in" crowd manages to prevail in most legislation. Given the social and political strains the war put on societies OTL, we'll assume it ratchets up a even more in GB. By 1919, Ireland is probably garrisoned by a lot more weary British troops that just want to go home.
Still, we'll assume that the hard liners win another khaki election and stay in power. Ireland will not be partitioned nor will it become a Dominion. In fact, the whole concept of dominion status is seen as anathema, which kills the Statue of Westminster. Now you have Canada, Australia and South Africa with their noses out of joint. Not irredeemably so but let's make sure that cooler heads do not prevail and the 1920's are a replay of the 1770's-attempts to coerce the empire together by political (not military, though, outside of Ireland-but occasional intemperate remarks are made to the effect of the possibility) force, along with economic policies (a resurrected form of Imperial Preference?) to "tie together the Empire" but seen as a way to channel free trade to/through (and for the benefit) of London. Of course, things only go from bad to worse with the Great Depression.
On the domestic front, since the Irish and continental wars were seen/fought as one, a lot of the regulations and wartime restrictions are still in place. (Maybe some form of DORA still in effect?) the General Strike probably gets much more violent as it will be seen/portrayed by the government as betraying the war effort (think the US during Vietnam). This results in the government calling another election that maintains its majority.
On the one hand, Parliament tries "get tough" legislation (but with no real way to enforce it) while, to try and drum up some goodwill both within and for the empire, the popular Prince of Wales is sent on a tour, with great success. The end of the 1920's sees another majority in the election as the economy finally looks better and the opposition is mostly in disarray.
With the background of an unending Irish guerrilla war (including contingents of Imperial troops), relations grow more strained as India talks of home rule and the government comes down hard (like mass arrests, imprisonments and long harsh sentences, including death) on Indian nationalists. This, says London to the Irish, is where talk of loosening Imperial ties will get you...but Canada, SA and Australia think it's aimed at them, too.
With the economic crisis of the depression, the government, though not in the majority, spends most of the early 30's in effective opposition, preventing any substantive changes by succeeding coalition governments, which dissolve frequently. Reinvigorated by the threat from Nazi German rearmament, they return to power around 1937-8 with a "See what happened the moment we weren't in charge?" campaign...
This, I think, may set enough of the stage for WWII to be fought with a rightwing nationalist government that still manages to piss off the rest of the empire enough to limit their war assistance to the minimum, given grudgingly, and with a view toward presenting London with a large bill (payable economically and politically) for services rendered at the end of the war. At the end, The government (which suspended elections for the duration of the war), finally goes off the deep end and suspends elections indefinitely, earns US public enmity with a final, savage, end to the "Irish Problem", and plunges into domestic economic chaos. Meanwhile the snubbed dominions decide that, if they are treated no differently than India, then they will act no differently. Between 1946-1950, Australia, India, New Zealand, South Africa and Canada, along with various minor dependencies, declare themselves to be dominions or independent.
Between the collapse of the empire and the economy, Scotland and possibly Wales decide to have a go at devolution. This is the straw that breaks the camel's back and everything is pulled home to put down the insurrections. Neither the Soviets nor the US have a way of intervening directly (but are willing to fund factions indirectly, lest the other one's possible allies take power) and the emerging Cold War puts both of their focus on Central Europe, anyway. The English Channel is no longer a moat to keep invaders out but to keep citizens in.
Cover, simmer for 30-40 years, and-voila!-the Hermit Kingdom of the Western Hemisphere. With the difference that they would still be on their first monarch.