It quickly became clear that while the chords of the song are pretty simple (the entire chorus is sung on an A chord!), the key to this one is rhythm, and that maintaining that rhythm on the uke while singing at the same time was not quite going to happen. I've also been wanting to enter into the world of multitracking, so this seemed a good chance to try it. (I also took the opportunity to buy a particular audio-editing package I've been wanting for some time, for all the good it ended up doing me.)
After many, many attempts, I got a decent ukulele track recorded, using the pickup built into my concert uke (his name is "Stitch"—the soprano is "Lilo"; you may detect a theme) to record direct to my PC. Then it was on to recording the vocal track. I got that in the can on the second attempt or so, only to discover that the microphone was on the fritz, and that much of it hadn't actually recorded. While trying to find a way of fixing that, I somehow rebooted the computer. Did I mention that I hadn't saved any files yet?
Fortunately, I had been relying on the fact that the microphone in question is unidirectional, and had been playing the uke track at full volume. And I'd been using my usual camera to record the whole thing... so the upshot is, the audio for most of the final product consists of the ukulele track being played through a set of cheap computer speakers, while I sing over it into a microphone that is essentially functioning as a prop.
I am frankly amazed that it turned out as well as it did. And at least I have finally justified the "Snuggie Kazooist" YouTube channel name...