Enterprise: The Good That Men Do established that United Earth is a
federated state, which means that every member polity gets to keep its own government to handle internal manners. There's no reason that United Earth can't have unitary ceremonial monarchies like the United Kingdom of Great Britain as member polities alongside federal parliamentary republics like the Federal Republic of Germany alongside unitary parliamentary republics like the State of Israel federal presidential republics like the United Mexican States alongside unitary presidential republics like the Republic of Chile.
I agree with that, I just found a bit weird that each constituency would have such different system of local government. It would be like having the Governor of Tennessee alongside with the King of New Jersey, the Glorious Leader of Texas and the Caliph of California.
For whatever it's worth, from 1776 to 1790, there was no Governor of Pennsylvania. Rather, the state was headed by the
President of the Supreme Executive Council of the Commonwealth of Pennsylvania (with the Council comprising the executive branch), and from 1776 to 1792 ten men served as
President of Delaware. And, mind you, George Washington first assumed office as President of the United States in 1789, and the other states were headed by governors. (Just think: If there had been a governor's convention in the White House, you would have had to specify who you meant if you were to shout out, "Mister President!")
So, really, the idea of a federal state's member polities have varying executive titles is not without precedent in real history.
More seriously, I was just wondering if, in such a system, they would keep the the ceremonial head of state (the King or Queen) along with the actual head of government (the Prime Minister of UK, for example), beside touristic reason. You don't see many Earls or Dukes having actual/ceremonial power over their former lands, nowadays.
I'm sure that within the larger United Earth, the Prime Minister of the United Kingdom would remain the head of government for the U.K.
Federation by Judith and Garfield Reeves-Stevens established that the Optimum movement had overthrown the Royal Family and transformed the United Kingdom into a republican dictatorship before World War III. Your mileage may vary on whether or not that information is still consistent with everything that's been established about 21st Century Earth since Star Trek: First Contact opened.
If memory serves,
Federation did establish the surviving royals had gone into exile in Canada, so the question then is, once the Optimum Movement was overthrown, were the royals invited back... or was there anything left for them to come back to?
*shrugs* It also begs the question of whether or not
Federation and its Optimum Movement should be seen as in continuity, in whole or in part, with what
Star Trek: First Contact, ENT, and
The Lost Era: The Sundered established about 21st Century Earth.
I also seem to recall that either Mack's A Time To duology or KRAD's Articles established that Buckingham Palace was used for Federation state events, which either means the Windsors had been kicked out, or maybe had moved into more modern digs by the 24th century.
Both, actually.
A Time to Kill featured Buckingham Palace being used to host a state dinner for Federation Councillors and the Federation President, and
Articles explained that the London Dinner was an annual event held for newly-elected and other select Councillors.
That doesn't necessarily mean that the British Monarch is no longer in residence in Buckingham Palace, though, or that the Monarchy has been abolished. Buckingham Palace is
big, and it's by no means outside of the realm of possibility that the Federation government uses the Palace by invitation of the Monarch.
My personal feeling is that they're gone, but I don't recall any evidence either way. Perhaps the royals all died during the bombings of World War III and the monarchy was done away with? If they are there still, I imagine they're even more ceremonial than they are now. 
