So I asked the local chain tyre place to quote for and then fit Winter tyres and they offered Michelin, which sounded fine.
They made a massive fuss when I wanted my summer tyres back afterwards, telling me I didn't need them which was weird, I kept telling them I'd had winters tyres for years on my previous car and always got my tyres back to store for next time.
I've now googled and they've fitted Michelin cross-climate 2 tyres, which are all-season, not winter tyres. I think they're supposed to be left on all year, but I've no idea if they'd be safe in a British heatwave, or if they're even good enough to act as winter tyres (we live at the top of a hill in Yorkshire, where the gritting is done by tractor).
The tyre place were terrible in almost every respect, so even if I asked them I doubt they'd give me a straight answer. Can anyone enlighten me please?