Thanks so much to tezza, Gibsaw and ZL2AOX for helping identify a couple of old computers.
Seeing as you did so well, how about a VIC-20 tape?
I remember it having about 10 games on it, and I remember playing a gamed called "Bomber", which was a version of Blitz, though I remember it being black on cyan, rather than a white background. It may have been around number 7 or 8? I can't remember anything else on it but I'm sure I'll recognise them.
I imagine it was a commercial (shareware? pirate?) compilation, as it had a proper printed cover, and I don't recall there being a picture on the cover. The inside cover had a list of all the games on it. It could have come with the computer - I doubt my father remembers where he bought it, but we would have been living in Timaru at the time.
Any ideas?