I don't know enough about how the British government works, and I'm
looking for recommendations on a good layman's book on British civics.
 Several titles, if possible.

It must include thinks like how the House of Commons and Lords works,
the "Constitution," the various parties, basic recent fiscal history,
etc.

Not too technical, but not so watered-down that a good encyclopedia
article wouldn't tell me the same info.  Something for the general
reader, preferably in a lively style.

Not sure about lively, but The Changing Constitution, edited by
Jeffrey Jowell, is standard issue to first year law students in the
UK. Search for it under ISBN 0199264392. It's probably more
intermediate than a beginner text for UK civics. I'm sure other
researchers will have other suggestions.