Re the Thames; downstream of Teddington Lock it's administered by the Port of London Authority (PLA) and you would need their permission to operate off it; upstream of Teddington Lock it's the Environment Agency, and you would need a river licence from them plus their permission to exceed the 8km/h speed limit.
Now I remember when Thorpe Park first started, they had a replica Supermarine S5 racer (Schneider Trophy type) and when it displayed at Farnborough, (late'70s) it operated from the long straight stretch of water used for the Henley Regatta!