My vet sends any animals that are not taken home to be buried by their owners to Cambridge Pet Crematorium, where they are cremated together, unless you make specific arrangements to have the ashes back, in which case they have to be done separately. You can collect the cat and take it to a private pet crematorium. Personally, although it is upsetting, I would find collecting ashes even more upsetting, and I certainly would not bury a cat as I move too often!
Just remember the good times, and look forward, there will be many times you think of things that your cat did, and I am sure you don't need ashes to remind you. Thats my opinion anyway.