Doesn’t come up very often here but I took an application from someone who has an umlaut over the second letter of their name. Would this follow a name with the same first two letters but no umlaut? That is my surmise.
Even the librarians can’t agree. I got as far as Copy and Pasting four different opinions from this group before I gave up. 
Just be glad the artist formerly known as Prince didn’t apply.
In a page about German grammar (strangely enough it’s a Finnish page in German, I found no German pages about the subject, go figure) I found that there are usually two accepted versions:
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Old version: Since an umlaut derives from the combination of the original letter + e (ae, oe, ue), they can be grouped as if they still were written like that.
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Newer version: Like you said, the word with the same two letters without umlaut comes first, then the one with umlaut.
I know both to be used, so I guess you can pick and choose 
Wow, just finished with Duck Duck Goose’s link and the consensus there is index it as if the umlaut isn’t there. Alternatively, rewrite the word as {umlauted vowel}+e and index accordingly.
My Langenscheidt’s German dictionary (pub. 1976) uses method 2)
It’s also the way I always learned it. The umlauted vowels are vowels in their own right.
True, version 2) should be correct in German, and thus applies to English too.
In Swedish and Finnish the alphabetical order is different, though, since these languages have ä and ö as independent letters and place them to the very end of alphabet, after z. Other languages probably have their own ways when indexing them.
*Umlaut, umlaut, doompadee doo
I have a follow-up question for you
Is it wrong to use the umlaut-thing
In English, when spelling “Hermann Goering”?
Or is it better that we be trying to
Use the umlaut like the Germans
Doompadee do?*
Sofa King–:D.
What about French and its trema? How do the French order their accent marks?
Hmm, second thread in a week or so about this. As I stated (sort of) in the other I’m a professional sorter and I disregard diacritics. The reason for this is that I think it is very unlikely that there will ever be a situation where the order has to be taken into concideration.
**
Would you care to divulge a link?
This is a subject that interrests me, so I would gladly read anything you have to say on the subject, BUT I cannot find the thread. Searching for “Floater + umlaut” gives nothing.
I find it fascinating that the letters are given different ordering in different languages, and I’d like to know more about where this convention came from.
After long last I found it. Here we are.
Ahh, Thank you!
Not that it really says much about the reasons for different alphabetical orders, but it does offer interresting insights to other ordering systems.
As for French, I just checked Le Petit Robert and, as expected, the word e’leve’ came before e’le`ve (sorry, but I don’t know how to make accented letters, so I have put the accents after the letter). The first place they differ is with the second e and they have put the accented on second.
I recall trying to find the city of Aarhus in the Jutland phone directory. It is, after all, the second largest city in Denmark and I just couldn’t find it. It turns out that it is indexed with A-ring (sometimes it is spelled with A-ring instead of double A) and that is the last letter of the alphabet and Aarhus was nearly the last city listed.
In a dictionary, where each word is supposed to be in a certain order, it makes sence to differentiate between characters with different diacritics (e, é, è, ë and what have you).
In real life outside dictionaries the problem would not arise very often as you compare two different strings and it would a be very rare case that they happen to be in the exact same place preceeded by an identical content.
To make it even more confusing, as was pointed out in the thread I mentioned, Danes, Norwegians and Swedes don’t put our special letters in the same order.