English names are pronounced the English way, just as you described. (Another example is Justin)
This is also true for most names from other languages. Just when such names contain consonants or vowels that are not used in German, they are replaced by sounds that are common in German.
German has a long tradition of importing words from any other languages, and very often people try to stay close to the original pronunciation. (This depends on whether the word entered German vocabulary as spoken word or as written word.) Words like the english Baby, or the french Chef are pronounced very similar to the original pronunciation, and we do the same with names.
The English Jute is one of the rare examples of words that came into the German vocabulary as a written word. It is pronounced like it was a German word ([ˈjuːtə], not [ʤuːt]), and most German native speakers are not even aware, that this is an English word (which itself has its origin in the Odia langage).
Also some brand names (like Colgate and Michelin) are pronounced as if they was German words, but only in Germany. Producers of TV-commercials decided to use German pronunciation for those names, and so people in Germany learned this kind of pronunciation. But those commercials was not broadcasted in Austria. Commercials produced for Austria used the original (i.e. English and French) pronunciation, and for this reason you can hear those original pronunciations when you listen to German native speakers form Austria.
Also interesting is the pronunciation of Donald Duck and Goofy before the 1980ies (at least in Austria, I have no knowledge about the situation in Germany or Switzerland). There was no spoken commercials for Disney Comics, and those both names was not translated into German (for example: Mickey Mouse and Scrooge McDuck was translated to Micky Maus and Dagobert Duck). English as a second language was not very common in those days, and so people pronounced the names with German rules: Duck rhymed to the english took (like in "he took a photo"), and Goofy had the same vowels that appear in office. But then more and more people learned English, and now only old people (65+) still use the German pronunciation for Disney Characters. The big majority now uses a pronunciation that is very close to the original one (just with the expectable German accent). (But Uncle Scrooge still is Onkel Dagobert.)