He watched as his six doppelgangers rummaged in the sacks, pulling out sets of clothes, putting on glasses, stuffing their own things away. (HP7, Ch. 4, p. 52)
As the quote from the other thread shows, in the American edition at least, there isn't an umlaut, and the word is treated as a normal English word with regard to pluralization and capitalization. That seems right to me. The word has been used in English for a fairly long time, since before the computer era, when special characters like umlauts became more easily available both for typesetting and for everyday use.
But it's also just a matter of familiarity. Since the umlaut isn't used in English, people tend to just omit it, even nowadays when they could find it on their computer; witness the current rash of uber- this and that. We've had discussions before about Dusseldorf and Zurich and so on. (Gewürztraminer may be a more limited demographic, BTW; it's not a wine like Chardonnay that everyone has heard of, so the people who know it exists at all may know it has an umlaut.)
For me personally, it's nice when someone thinks to include an umlaut, and I expect that the trend to add them more often will continue, now that it's easier. But it doesn't bother me when they're omitted in English texts, because it's not really a mistake. We are not required to substitute ue if we don't have ü on our keyboards. The same is true, BTW, for words from other languages, like facade, jalapeno, Dvorak ...
The decision about whether to use a capital letter depends on whether the word is still regarded as a foreign word, as opposed to an imported word. Words that have been imported into the English language and are fairly common, like kindergarten, are listed in dictionaries, lowercased, and treated as English words. Words that are unfamiliar to many English speakers are more often kept in italics and spelled and capitalized as in the original language. I would personally put Weltschmerz in that category, and doppelganger in the more familiar category. But it's not an absolute, either-or thing, more a matter of feel, and dictionary spellings may vary. So when you want to use less common words of German origin in English, how you write them is to some extent a matter of feel for your audience, and of personal preference.
But when someone else uses a German word and doesn't put it in italics, you can assume it's considered English and treated as such. Like the way Babys is considered German. (-:
I was kind of impressed, in a very small way (okay, my expectations for HP7 were pretty low), that there were actually a couple of sentences in German in HP7, at least correctly spelled, and even one word with an ß:
'Er wohnt hier nicht mehr!' she cried, shaking her head. 'He no live here! He no live here! I know him not!' ...
'Das weiß ich nicht! He move! I know not, I know not!' (Ch. 12, pp. 232-33)
Though, as someone said in the Rowling/colon thread, the person being looked for is called Gregorovitch and from the context it sounds like the scene is meant to have taken place somewhere in Eastern Europe. And, come to that, 'He no live here' actually sounds more like Spanglish. Oh well. Details. (-;
@Russisch Brot: Hope you feel better soon.