You might try calling, although that might just go to central customer service support.
I've made appointments in person at an Apple Store where I didn't need to provide a serial number or an Apple ID. That might not be convenient if it's a long trip that you don't want to make more often than you need, although there might be the possibility that they'll allow a walk-in evaluation. I tried online and there doesn't appear to be any way to make an appointment without an Apple ID.
It should be possible to get it repaired in the UK, since the part should be universal regardless of country.
I tried again, and (at least in the United States) I see a way around having to provide a serial number. I started here (I'm hoping this is the same for you):
Mac Repair & Service - Apple Support
Click on "Get Service".
Click on "Continue without device".
Click on "Physical or liquid damage".
Click on "Cracked screen or display".
Click on "Continue".
Click on "Find a location".
Click on "Continue without device".
Then pick a location and then a time. Some of the locations may not be Apple Stores but AASPs. It should know your approximate location, but if it doesn't it might prompt you for your location.
Pick a time.
The rest would be them asking for a phone number and I believe your email contact information will be your email used for your Apple ID.
Good luck.