Ouch, Bloss! Those look painful! Sorry, no advice. I got a chilblain on one big toe a few years ago, GP prescribed an antibiotic which did bugger all of course. I was a little bit annoyed about that but I guessed he hadn't much experience of chilblains, and neither had I!
Anyway, an electric underblanket on my bed sorted the problem out. For some reason my feet, usually too hot in the evenings, go into super chill mode when I go to bed.
I hope you find something that helps. Mitts in bed?