"To Anachreon In Heaven"... The tune appears to have been a collaborative effort by members of the society. It's been claimed that it's a Carolan tune.

That's quite credible - it's a good enough tune, and it has that kind of feeling about it. But I don't know if there's any evidence for the claim.
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As pointed out, lots of settings of Kipling songs prior to Peter Bellamy - but I'd say his tunes knocked the rest into a cocked hat, and felt uncannily right.