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
I swam in the river Jordan
I played the game Half-life
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.