So the mystery is (almost) solved. Gretel Parker has emailed in a copy of an article in the August 1955 Country Fair, celebrating John Hanna’s fiftieth cover illustration. It’s nice to know he produced at least fifty – we’ve only got five, so there are obviously plenty more to discover.
It also seems he is the same John Hanna cited in the link that Adam posted in the comments – an Australian who arrived in London in 1947 and made his living as a commercial artist and cartoonist. So we now have the full biog and background. Only remaining mystery is why his works aren’t more widely available, or indeed available at all. Reading the article, it seems he was extremely popular in his time. Strange the way people can drop off the radar like that, leaving barely a trace on Google. There seems to be a fan site for everything these days, so it's a surprise when you find a gap like this. Or maybe it’s just the modern mindset: if it’s not on Google, it doesn’t exist.
Anyway, thanks to those who have put us out of our ignorance. Look forward to hunting down more cover illustrations in years to come.