Strangely, it is idiomatic (at least in British English) in connection with plastic (cosmetic) surgery.
"Work done" is usually work that you have had someone else do for you, for example, "I'm having my kitchen done" is an idiomatic way of saying in BrEng that you are having your kitchen refurbished.
Cosmetic surgery is usually something that a person elects to have, and at their own cost, which is perhaps why it is spoken of in similar terms. For example, "she's had her nose done" is an idiomatic way of saying someone has had cosmetic surgery on their nose.
When it comes to medical procedures, it certainly is nowhere near as common, but it depends on the kind of surgery. For example, prosthetic hip replacement is quite common, and I have heard this described as "I'm having my hips done". Something like a tonsillectomy, for example, would not be referred to in this way. We would just say "I'm having my tonsils out".