Eventually I used MelodyCatcher and it told me it was the same as this midi file. Which would be more useful if that file had a meaningful name.
At which point I went on to IM and started pinging people I knew who might be able to help. Went through a John Wilson (recognised it, didn't know the name), Lizzie, Ari and Erin (couldn't quite place it) and Hannah (also didn't know) - and apparently it drove them crazy too, to the point where Ari called her brother to ask, Lizzie called her mother and sang it at her, and _she_ sang it to her sister to see if they could track it down, before resorting to SoundHound for suggestions, which came up with the correct answer as a "maybe", so they YouTubed it to see if it was right. And they would have won it too if Hannah hadn't had a brainwave (and a bit of googling) and beaten them to it by 47 seconds.
See if you know, and I'll stick the answer under a cut to make it more fun.
It's the Turkish March (Turkish Rondo/ Alla Turca) by Mozart. Or as Julie said, once I played it on Spotify, "The music from Lemmings".
Original post on Dreamwidth - there are comments there.