Is the fingering on a clarinet the same as on a recorder?

Not according to http://www.paythepiper.co.uk/switching.htm

No, but very close. The fingering on a recorder would be closer to that of a saxophone.

Flute, Recorder, Saxophone, Oboe and Clarinet all have different fingering from each other, but the Clarinet is the most different from all the others. It's not difficult to pick up, though. The hardest thing about the clarinet is that the upper register (high notes) have a different fingering from the lower register (low notes). In most instruments, the fingerings in different octaves are basically the same as each other. This is not the case in the clarinet.

Thank you everyone. My children are of an age to start learning a musical instrument - so we have been talking recorder (GBP 8) vs. clarinet (GBP 400).
It seems the only strictly transferrable skill is the ability to read music. But I want some proof of staying power! We have settled on, they will start to learn the recorder, and if, this time next year, they can read music and have proved they will keep at it, we will pay for the instrument of their choice (and lessons). Well, as long as it is offered at school, not a state-of-the-art electronic piano!