Having seen dogs with only three legs both in person and on tv (on vet-type programmes), they seem to adapt remarkably well and remain happy and still run around. If the dog still has a good quality of life and is not suffering, I'm sure it will be fine. Obviously you wouldn't have it done if it would prolong the suffering..
I guess it would depend on the age of a dog, an aged dog would probably find it harder to cope. Also I would think that some breeds have an easier time adapting that others, due to body proportions etc.