What (if any) is the "official" name for this kind of song?

They’re called cumulative songs

I thought I knew a different word for them, but as soon as I saw @thorninmud‘s answer, it drove any other term right out of my head. So I’m not sure if there’s actually another expression or not. But you’re certainly safe with that one.

I was thinking of Rounds but that’s slightly different. I am with @Jeruba that there is another word for them too as I’ve not come across cumulative, though it makes good sense.

Also known as “Formula Songs” – for those interested, an article