The speed of the Gulf Stream, at the best places, is about 1 m/s.
Trough a surface of 1m2 perpendicular to the flow direction, that makes a flow of 1m3 per second. 1 ton water per second.
A mass of 1 ton water moving at a velocity of 1 meter per second has a kinetic energy of 500 joules (Ek = 1/2 m v2).
Thus, a watermill of 1m2 surface, placed into the flow of the Gulf Stream, would receive from the water a power of 500 Watts, that is one half kilowatt (P = 1/2 rho S v2) (rho is the density of water: 1000 kg/m3).
Let's assume a final efficiency of 20%, that makes 100 W per m2 watermill.
Thus, a big underwater flyer, anchored on the bottom of the ocean, carrying water turbines, will be able to deliver the same power as a conventional power plant. A flyer with a diameter of 120 meters would deliver a power of 1 Megawatt.
The flyer must not necessarily be anchored: the Gulf Stream flows in both directions, it is just a matter of depth. So you can use a combination of two flyers latched together trough a long rope. One flyer near the surface, the other near the bottom. They will remain at the same place and produce huge quantities of electric energy.
I don't know if it would be difficult to transport that electric energy to the continent. If it is, then let's use it where the flyer is. An industrial site could be placed in the neighbourhood of the power plant, floating in the middle of the ocean. Ships would bring the raw materials and carry away the finished products. Some raw materials can be furnished by the ocean himself.
Why not power that way the oceanic cities of the future?
The big advantage of the Gulf Stream, against windmills and solar plants on earth, is that the power available will be very steady and reliable. Just one thing: don't forget the fish and the inhabitants of England. There should be not too much flyers placed into the stream, because they will partially blockade it, and the turbines should not turn too fast, in order not to hurt the fish.