Although Hawkeye doesn't actually claim that his wrist is broken, he does say that he can't move it, and a plaster cast is used to immobilize the bones. Thus, the severity of the injury would prohibit him from performing surgery for several weeks at minimum. Regardless of whether or not it was broken, with Hawkeye unable to function as a surgeon, I Corps certainly would have transferred him out of the 4077th and permanently replaced him with a new surgeon, either eventually sending him to a new unit when he recovered or rotating him home.
The wounded Batallion Aid surgeon needs immediate surgery (shocky, with a collapsed lung, along with chest and belly wounds), yet Hawkeye and Dr. Borelli are shown operating on him each wearing a forearm-length plaster cast after being knocked over by an artillery burst. Setting a wet plaster cast and letting it dry is a lengthy process -- certainly time that they couldn't spare in saving the life of the badly wounded surgeon. In reality, the wounded surgeon likely would have died while Hawkeye and Borelli waited for their casts to dry.