Man, I wish I'd known that I would have to copy the files off before removing the hard drive. Who'd have thought that the drive wouldn't be readable connected to another device!
Problem with the "just copy files onto your internal hard drive" is that when you have Time Machine backup files for several machines, it's very likely you won't have a hard drive big enough to copy the files onto... Catch-22!
So I've just started backups from scratch... Once it's done successfully then I'll wipe the old drive and use it for something else.