In British English, it would be River Thames, River Jordan, etc, (as opposed to AmE requiring “River” after the name). So:
I swam in River Jordan.
But, what happens to river’s capitalisation when the is added?
I swam in the River Jordan
OR
I swam in the river Jordan
as in
I played the game Half-life
Edit
Damn! I just came across a PDF on Chicago Manual of Style. Agreed it's for AmE, but the first page itself talks about “the river Elbe” having a lowercase “R” because it’s “added by way of description”. Hence, my question stands. Does something similar apply to BrE or do we stick to capitalised “River” in all situations as long as it’s followed by a proper noun.
Best Answer
When you name the river, you capitalise river, so it would be:
but without the name:
The presence of 'the' is less relevant than the presence of the name of the feature. There are similar conventions with other natural features, for example mountains, e.g. Mount Everest.