Stagger, Yes, a southbound track does converge with the river just to the north of Vauxhall and the pilot would have the river converging on his left. However, unless a 90 degree right turn is made immediately at the VRP, the aircraft will cross the river because of the bend just to the south of it's location. Obviously, any aircraft has a turning radius commensurate with speed and a delay of only a few seconds would make a difference to crossing the river or not.
Don't know if you've seen one, but it's obvious on the CAA London helicopter routes chart. Immediately south of Vauxhall bridge VRP, almost on the south bank, lies the crane in question.
A helicopter already established on the river, having joined H4 further east, would be slightly better placed with regard to making the turn, but not by much.