I think this is just a case of forming words from two parts (Komposition).
In the context of rivers there are indeed lots of variations all involving things usually to be found near rivers. Examples of such things may include -wiesen, -niederungen, -aue, -wehr, -mündung, -delta, -promenade, -deich, -tal, -ufer, -strand. (I'm sorry I can't provide translation for all of these.) Some made it even to city names, e.g., Travemünde, Paderborn.
Also you may find terms describing events or buildings or professions combined with names of rivers or seas or even oceans, e.g., Elbphilharmonie, Elbhochwasser, Elbvertiefung, Nordseefischer, Atlantiküberquerung.
As you can see, these are sometimes formed with a shortened form of the name of the water (Elb- vs. Elbe), and they appear a bit off when formed with names of two words, though people will still occasionally do it (e.g., Steinhuder Meerufer). And I can't remember a case where I've noticed a compound containing the english name of the river.
Also, most of these terms are known only in the region they appear, so most people may not recognize foreign sites at once (Podelta), others are however more common (Amazonasdelta, Nildelta).
In conclusion, I don't know what river you are actually talking about.
If it's original name is Red River, one might potentially say something like:
Ich war im Urlaub am Red-River-Strand. (If there is such a thing)
But especially in written language you would most likely rather put it this way:
Ich war im Urlaub am Ufer des Red River.
If it's this Red River instead (original name in a language other than English), one would probably write and say the latter, but with the german name.
... am Ufer des Roten Flusses.
Unless you want to show-off you knowing the name in the local language (Vietnamese/Chinese?), in which case you would use that name.
In any case you should take care to provide more context to make sure people know what river you are actually talking about.