I'd use either.

Fun fact: did you know that Avon is a very common name for a river in England. Why? It comes from the Celtic word 'afon' meaning, literally, 'river'. So, all the Avon rivers called River Avon are bilingually named River River. Depending on the purpose, "Bristol is situated on the River Avon." is all you need. The rest is self-evident or superfluous in ceryain circumstances. We wouldn't need, for example, to mention that Chongqing is situated on the Yangste River, CHINA. If you have heard of Chongqing, you know it's in China, already. Context is everything.

If you write it you don’t use the "in" but if you tell someone about Bristol you would say .... on the river Avon in England.