How do wing tip vortices interact with the airflow on an wing with winglets?
Winglets do not change the laws of physics. In particular, they do not change the flow around a wing such that there is no up- or downwash. All they do is to involve a little more air into the creation of lift such that the vortex strength is slightly reduced.

With winglets, wingtip vortices now form at the tip of the winglet instead of at the wingtip. The wake behind the wing still rolls up and creates the usual wake vortices into which the wingtip vortices are absorbed. In order to create the same lift with a slightly less powerful vortex, the distance between both vortices is increased slightly. Actually, in order to support the winglet's weight, the aircraft needs to create slightly more lift with winglets than without, which reduces their benefit.