The current position of the law is settled in that no general public right to navigate in non tidal
rivers exists in England and Wales. While the public has the right of navigation in tidal waters (e.g. Gann v Free Fishers of Whitstable (1865) 11 H.L.Cas; Blundell v Caterall (1821) 5B & Ald. 268), this depends on the presumption of the Crown's ownership of the land beneath the water. This presumption is rebuttable and there are some instances where the tidal riverbed is under private ownership.
The presumption of rights of navigation on tidal rivers contrasts with the very limited right on non-tidal rivers. The default position is that there is no such general right of navigation. Above the flow of tide the land beneath a river or stream is privately owned so that while the public can acquire navigational rights over such waters they cannot have them as of right. It has been held that rights of navigation on inland waterways are not analogous to rights of way on land (Wills' Trustees v Cairngorm Canoeing and Sailing School (1976) SLT 162 and AG
ex rel Yorkshire Derwent Trust and Malton Town Council v Brotherton  1 All ER 230).
My impression was that because it was in Scotland it doesn't apply to us south of border, although it's been quoted in a case in Yorkshire.