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So, for anyone interested, here are my findings:

1. Third wheel is (from my very unscientific polling) more than twice as common as fifth wheel, but both are in use.
2. Fifth wheel is a type of camper (particularly to Americans.)
3. Brits say "Gooseberry," which is delightful.
4. The original phrase seems to be "like a fifth wheel on a wagon," and "third wheel" was a misunderstanding of the importance of the number of people rather than the number of wheels, but has become the dominant form.