We know of only two rivers by name: the Mersey River and the Wye River. A third, the Otter River, is assumed to be near the Burrow and therefore serve as the namesake of the nearby town – Ottery St. Catchpole. Harry, Ron, and Hermione also spend nights by the banks of unknown rivers while searching for Horcruxes, though they are never named (DH22).
There are certainly other rivers – the Brockdale Bridge spanned a river somewhere, and the Leaky Cauldron is very close to the River Thames in London – but these have never been specifically mentioned.
See also: lochs, oceans, seas