I'm in the uk and have a sabacous cyst in a private place. Nhs refuses to treat because it's cosmetic. What can I say to get them to treat me?

It is a pathology. Sebaceous cysts are accumulations of skin oil due to failure of ducts to get it to the outside skin. Body reacts as if a foreign material and walls it off with fibrous tissue forming a pseudocyst. This requires opening and drainage, to include the gland tissue causing it. May become infected and require bigger surgery. Not cosmetic.
Painful or bleeding? Is it painful or bleeding? Does it seem infected, draining pus? Does it interfere with urination or sexual intercourse? Is it rapidly growing which might be a sign that it might not be a cyst but a tumor or cancer.
If it is infected. If it is large and infected, you should not have any problem getting treated. But, if it is small and asymptomatic you may have to do lot of explaining.