Final update:
The verification codes are arriving around 20 minutes after I request them, but the code expires after 10.
I’m convinced the issue is with Hotmail rather than Nintendo, as others have just seamlessly got their codes and logged in.
It’s all good though, I’ve managed to preorder the consoles and pro controllers from other retailers, and @CTD kindly got me a GameCube controller and SD from Ninty direct.