The UK doesn't have one central pet-emergency phone number the way it does for human emergencies (999). Instead, most regular vet practices either run their own out-of-hours rota or refer after-hours calls to a dedicated emergency provider covering the local area, and that referral number is usually on the practice's own answering message.
If you're registered at a practice for the trip, or staying somewhere with a local vet recommendation, save that number as soon as you arrive rather than searching for one mid-emergency. The Royal College of Veterinary Surgeons can also help locate a registered vet in the area if you're starting from nothing.
UK veterinary care is generally excellent but not cheap, and out-of-hours or emergency consultations typically cost noticeably more than a standard daytime appointment. Have a payment method ready that can handle an unplanned, larger charge, since most practices expect payment at the time of treatment.