Is there a reason why Americans usually place the word "river" following the name of a waterway, for example, the Ohio River or Rio Grande River, while the British often place the word river first, as in the River Thames or River Ouse?

It's seems to be historical, for reasons unknown:

Once upon a time, river names in English usually included the word "of." So instead of "River Jordan" (in modern British usage) or "Jordan River" (in American usage), you would have found "River of Jordan" (written something like "rywere of Iordane"). Many of the Oxford English Dictionary's earliest citations for names of rivers, dating from the late 1300s, include "of." Chaucer in 1395, for example, wrote of "the ryuer of Gysen." This practice of including "of" in river names, the OED says, wasn't the only way of naming rivers, but it was "the predominant style before the late 17th cent." At that point, "of" began to drop out of river names, and British and American practices started to diverge. In proper names, the word "river" commonly came first in Britain, but last in the American Colonies. In other words, most English speakers simply dropped "of," but Americans reversed the word order as well ... But we haven't addressed the question "Why?" Why does usage differ in Britain and America? Why did the Colonists prefer "James River" and "Charles River" to the reverse? We can't answer that. But certainly the style adopted by the Colonists wasn't unknown in the mother country.

https://www.grammarphobia.com/blog/2012/05/rivers.html

In the UK we do have some with River after the name, eg the New Bedford River and the Old Bedford River https://en.wikipedia.org/wiki/Old_Bedford_River but those two were named after the person responsible for them - the Duke of Bedford. We also have at least one river that does not have river in its name - The Birket, which is a small tributary of the River Mersey, but which does have the River Fender as a tributary itself https://en.wikipedia.org/wiki/The_Birket (Streams, brooks, burns, etc do not count as rivers.)

I agree with gtho4's answer, but I also want to note that Americans would not refer to the "Rio Grande River", as the question puts it. "Rio" means "river" in Spanish, so the river is just called the Rio Grande, even in English. Spanish river names do get Anglicized in some cases -- "Rio Colorado" became "Colorado River", for example -- but the Rio Grande is too famous (and its Anglicized name, "Grand River", too common among US rivers, including rivers in Ohio, Michigan, Missouri, South Dakota, and Oklahoma, among others) for that to happen.

..and of course, we Brits still refer to the River Avon, which, as Avon comes from the Welsh word 'afon', meaning river, making the River River.

As there are three major rivers called Avon in England, we tend to qualify them as the Bristol Avon, the Hampshire Avon and the Warwickshire (or sometimes Shakespeare's) Avon, and tend not to append "river" to any of them.