I'm not disagreeing with the rest of your point - after all this is a plausibility check. But can you just explain why they'd have to give up their claims?
Consider the War of the Spanish Succession. The prospect of a Bourbon sitting on the thrones of Spain and France was enough to plunge Europe (and its colonial holdings) into a bloody war. The threat was just too great. The peace settlement eventually ensured that the lines could never be united.
Crucially, this was in an age where dynastic politics was still the driving force of European politics. Furthermore, the domestic view of monarchy is completely different. Louis XIV was King of France, not King of the French. A century later, when nationalism had overtaken monarchism? There's no chance the general public in either country would accept a foreign monarch who doesn't make a public show of renouncing their previous allegiance. This is a century where Prince Alfred, a man who would have struggled to be more German had he invented a more efficient sausage, used to dress up in a kilt to show what a wonderfully authentic Laird he was at Balmoral. This wasn't just for fun; there were plenty of monarchs who came to European thrones from other countries in this period but all the ones with half a brain leaned heavily on the national language and national dress of their new home. You can't put romantic nationalism back in the bottle.
Now imagine: the two greatest powers of Europe are going to unite. They will, once this union is cemented, control
over a third of the globe. Their military, naval and industrial capacity will equal, and probably supercede, all the other European Great Powers put together. In the City of London, it also possesses the financial capital of the world, plus the not inconsiderable German markets. Oh, and its scientific advantage from the combined weight of the British and German research establishments will be crushing.
It threatens the vital interests of all those other Great Powers in every traditional area of contention.
Africa? Cape to Cairo is suddenly on the table. France can no longer hope for British support to keep the Kaiser out of Morocco.The Boer Republics have now lost all hope for German backing. The proposal (seriously considered OTL) for Germany and Britain to carve up Portugal's colonies is now almost certain to go through. The race to secure the territories on the Upper Nile is suddenly vastly more advantageous to the British, as they can now mobilise out of German East Africa as well. All this will not, to put it mildly, be well received by France, Belgium, Italy or Portugal.
The Middle East? Any chance France and Russia had of getting concessions out of the Ottomans is gone. Between Suez and a Berlin-to-Baghdad railroad link, all exports between Constantinople and the Trucial States is going to be running on Anglo-German rail.
South America? Venezuela's suddenly going to be facing a united British-German front in the border and debt disputes. This won't play well with Washington, especially since the Southern Cone is
already in the unofficial British empire.
In Asia? The Anglo-Japanese alliance just stopped being relevant, now that the German Pacific Fleet is linked to the Royal Navy. That's going to leave Tokyo faced with the choice of becoming a client state of the Union, or a tenuous ally of France (and thus possibly a client state of Russia.) Plus, the Union is now unquestionably the dominant foreign power in China which means that all the careful turn of the century efforts to build a situation tolerable for the various Western powers has been wasted.
Oh, and to make things more fun: by uniting with Germany, Britain is making it clear to its Australian colonies that it's officially overruling all their long cherished ambitions in New Guinea, the Solomons and the South Pacific. That's not going to be received well.
Speaking of the Pacific, this also flips the Samoan dispute against the Americans. Again, won't be received well. They're also not going to be fond of the potential implications in Venezuela.
So, to summarise: even leaving aside the fact that the Union won't be accepted by nationalists in either Empire, it will have the unique historical effect of making enemies (all at once!) of France, Russia, the USA, Japan, Belgium. Portugal and Italy. Oh, and given the general destabilising effects in Europe, it won't go down well with Austria-Hungary either. Probably the Dutch, too.
You're also right to discuss the opposition of nationalists. Some potential counters: the timeline's original author assumes a chain of events that makes William II a more popular figure due to being seen as compassionate. Do you judge that as sufficient force to outdo the nationalist argument? Also, would there be a place for nationalist revisionism? That is, stressing the Anglo-Saxon heritage as being part of a wider German family?
Which leads to the Irish Question, where I completely agree with your assessment. It's a very good point against the scenario, although excellent fodder for conflict were a TL to be written - along with unrest from Wales and Scotland in such a scenario, if Anglo-Saxon heritage was used to justify the link at the expense of 'Celtic' identities.
Nationalist revisionism takes time. I've read far too much Victorian racial science for my tastes (it's featuring quite heavily in the background reading of my Phd,) and there was certainly a strong line of thought that the Teutonic and Anglo-Saxon 'races' were linked. That didn't stop war scares in Samoa in the 1890s or in the decade leading up to 1914. Given how unpopular this union is going to be, I don't see how it lasts long enough for people's identity to change, and that's assuming an incredibly successful propaganda campaign.
Also, remember that
this is Wilhelm II. He could be charming- in the short term. Anything longer than that, and he'll do what he always did: insult everyone and anyone whose good opinion he needed. The Kaiser thought he had a gift for personal diplomacy. He really, really didn't.
And even if he's incredibly charming: he needs to charm-
1. A German establishment with no interest in a union.
2. A British establishment with no interest in a union.
3. The voting middle classes in both countries.
4. The working classes in both countries, who were certainly proud of their monarchs but were also deeply nationalistic and had no desire to join the other nation. This is also an age of mass worker mobilisation, so you can't write off the populace as a serious political actor in their own right.
5. The
entire rest of the world.
He also has to do this starting from a position where essentially no other person in a position of influence shares his dreams.
Joseph Chamberlain made no real headway- none- in Imperial Federation. That was a project which had
vastly more support at both the popular and elite level than this proposed union, and was not anywhere near as threatening to the other Great Powers. It had the advantages of a shared language, a shared history, a shared culture, and a political and economic establishment that was far more closely linked than an Anglo-German union. And. It. Got. Nowhere.
I'm sorry, but there's no way in hell this is going to happen.