There are plenty of river ports/quays with power and water hookups. If you want to houseboat it, you really only need to move when you want to dump the septic tank (and in some areas, not even that).

The "good" spots are highly desirable and usually privately owned (or on a very long-term lease), and you often need to buy a boat to get the mooring spot beneath it. (A buddy of mine in London ended up with 3 boats while upgrading his mooring spot.)

Another options is to vagabond it, where you move your boat to a new area every week or two -- this is usually unserviced, so it's more like camping, but lets you use unofficial spots and is free.

(The last option is to find goofy loopholes. E.g., in some areas of the Thames, you can create a nesting habitat for an endangered bird on your boat, and then once the bird settles in, they can't force you to move, since doing so would destroy an important habitat. Sounds stupid but a surprising number of permanent city boats are there on some sort of loophole.)