Forgive me if this seems vague, but this is mainly looking at the Germanic languages. Proto-Germanic probably used an alveolar of some sort, most likely a trill.
In terms of Modern Germanic languages, all of the North Germanic languages utilize the /ʀ/ with the exception of Danish which uses a different uvular phoneme I can't quite describe (in large part because I still struggle with its pronunciation,) that I am aware of all. So because I don't know of any North Ger. language I don't know about, I cannot quite say of any that don't use a uvular sound other than Icelandic.
Of the West Germanic langauges, English's <r> in some accents remained /r/ but /r̥/ was lost completely and it was also shortened into /ɾ/ in others and the most dramatic change afterwise was the development of /ɹ~ɻ~ɹˤ~ɻˤ/ before the time of Shakespear and massive colonial expansion, which preserved this variation in colonies that would later develop into such countries as America, South Africa, Australia, and New Zealand, and from there, many dialects began to use the <r> as a length marker on vowels, prefering to drop it, except in initial positions and in consonant clusters.
Dutch and Frisian kept the trilled <r> exclusively, unless I am unaware of some obscure accent, and German's split with most of Germany trading to the voiced and unvoiced uvular trills, and some southern and rural parts and other german speaking countries, like Austria, using a mixture of the two either or using solely the alveolar trills.
Are there any good guess as to why this phenomenon occurred in the Germanic languages all?
I would assume that it might could have something to do with the popularity of the French language or possibly some sort of celts that were involved with Germans, but for the rest, I am afraid I can't guess.
Sorry if this is a bit long and is a brief history on the English <r>
