There are various scenarios - I've seen them all.
Hypothetical situation - dog hit by a car, leg broken, complex fracture.
Hypothetical £5k bill for initial stabilisation, x-rays and referral to specialist for surgical repair with pins and plates. Gold standard option, as close to full recovery as possible.
Can't afford that? Cheaper option would be to amputate the leg.
Can't afford that? Could try charities to see if they can contribute towards the bill. The RSPCA where I am often cover up to 25% if eligible, with the condition that the dog is neutered at the same time if they aren't already.
Can't afford amputation with charity assistance? Euthanasia. Yes vets have an obligation to provide emergency treatment and to prevent suffering, euthanasia is a viable option for that.
In other scenarios, I've seen the animal signed over to the vet, who will fix the animal up at their own expense and either keep/rehome. There's a reason why vet surgery employees usually have various broken pets! In other cases, pets have been signed over to rescue centres who have covered the treatment and then rehomed.
Most practices legally can't take payment plans as they don't have a consumer credit license. If you have a bill that you don't pay, they will pass it over to debt collectors.