Some English speaking people who don't speak German don't understand the meaning of the dots on a, o and u, but they can see, that those dots are typical for German language. German language also often is perceived as sounding hard, and therefore in the 1970ies some heavy metal bands began to use umlauts in their names because those umlauts gave their names an even harder image:
- Blue Öyster Cult
- Mötley Crüe
This kind of usage is called metal umlauts or röck döts, and those metal umlauts made german umlauts popular even to people who was not fans of heavy metal music. So many English native speakers, who have no idea of german language may think:
If it is German, it must have umlauts.
I think Apfelstrudel was known in USA even before Arnold Schwarzenegger, but it was him (also in the 1970ies and 80ies) who made Apfelstrudel really popular in USA (baked by his mom), and he came from a German speaking country (Austria) and has a heavy German accent.
Soon the German word Apfelstrudel turned into the English-German mixture apple Strudel, but instead of keeping the first letter of Strudel in upper case (which in fact really is very typical for German language), they used a lowercase s, but added dots to the u to make it look more German. And so the English röck-döts word
was created. When I search for "strüdel -strudel" today, Google reports 22.300 results.
So, this special spelling is neither correct German, nor is it correct English. But you will find it relatively often, but only on English websites.