It may be that your first admission is as an emergency with renal colic (stone pain) and that can be managed by any urologist.
However longer term management of your stones and cystinuria should be by an endourologist or a general urologist with a special interest in stones. Most hospitals will either have a urology department or have access to one. Some units may not have a specific person with an interest in stone disease and you may have to travel if you wish to see one. Your GP or local hospital should be able to advise you further.
Ideally medical management and monitoring should be in a specialist clinic with access to:
Urologist
Renal Physician
Dietitian
Availability of a nurse specialist/ practitioner