For the longest time ever I assumed these are two different places and was very confused about never knowing where Nuremberg is. Recently I found out that Nuremberg is the English form for Nürnberg.
Is there a known first documented mention or some other historical evidence which may point to the purpose of this change?
The English spelling reflects an older form of the name, as in mediaeval Latin Nuremberga, or Norimberga. In the local Frankish dialect the town in still called Närmberch. So really your question should have been "What is purpose of changing Nuremberga to Nürnberg?"
The original name of the town was Noremberg, English retained the 3 syllables after the name was reduced to 2 in German. This bit from the 1770's ( The Negotiators Magazine) describes money weights and measures from Noremberg:
The Coins of Noremberg. One Rixdollar is 1 1/2 Gould, 22 1/2 Batzen, or 3o Imperial Grosses, or 9o Cruitzers, or 4 f. 6 d. Sterling.
Sometimes there is no explaining why something is spelt a certain way in English. The same may be said of German in the Middle Ages. Not all cities are old enough or important enough to gain confused spelling and form.
Many world cities are at variance as to spelling with their local names in English.
There are many cities that have a different name in other languages from the one used by the locals: Cologne, Munich, Vienna, Lucerne, for example, in German-speaking countries. More exonyms were used in the past in English, like Mayence or Frankfort, but now there is a tendency to use the native names in most cases.
Some cities have many different names. Liège in Belgium can be hard to find following the road signs if you don't know that it is also called Luik in Dutch and Lüttich in German.
