No, he’s correct about Storm Shadow, this is from back in July:

https://www.independent.co.uk/news/uk/home-news/storm-shadow-missile-british-russia-ukraine-b2577953.html

I thought this had been the situation for a while. Didn’t they publicly back this very briefly before quickly walking it back?

Yes, Starmer said it back in early July. It sounds like the UK then looked to get agreement from the US and France before proceeding and the US have blocked it since then.