No, sorry about your misfortune, but use your passport.
Lot of countries have an ID card, but you're British and you live in Britian, so you can't get one.
There are alternative forms of ID for proving your age in the UK, like the "Proof of Age Standards Scheme", but I doubt those have any traction outside of the UK. Maybe you could use your shotgun certificate if you have one. I imagine some places are strict about these things, and some will take anything official looking with a name, photo and date of birth.
The thing about all these IDs though (including an ID card), is that it would take just as much or more time and hassle to get them as to replace your driving licence. There is no "emergency form of ID" except in the case where you actually do lose your passport.