My understanding is as follows. Is this universally agreed?
The OED sense 2a of surgery explains its use to describe the room where a doctor sees his patients. The OED gives no indication that this sense is exclusively used in Britain.
2a. The room or office in a general practitioner's house or a health centre where patients are seen and treatment is prescribed; the regular session at which a doctor receives patients for consultation in his surgery.
Nowadays where GPs work in group practices the individual rooms used by each are rarely, in my experience called surgeries. We refer to the entire building where our doctors practice as the surgery. "When you go to the village, could you call in at the surgery and collect my prescription".
As we sit in our doctors' waiting room, when our turn arrives an electronic sign with the patient's name on says 'please go to room 5'. Hence the place where you encounter the doctor is not strictly a surgery, but nowadays a room within a surgery. It is historically known as a consulting room.
As the OED explains, and as can be seen from the OED definition, a surgery is also a session during which the doctors are available to see patients. e.g. morning surgery starts at 8.00am; afternoon surgery at 3.00pm.
The OED also goes on to explain how people such as Members of Parliament, Accountants etc hold surgeries, borrowing the word from the medical profession.
1846 Bentley's Misc. June 549 A small den [Dr. Faunce] called ‘the surgery’.
1862 M. E. Braddon Lady Audley's Secret III. vii. 200 The door of the little surgery was ajar... The surgeon was standing at the mahogany counter, mixing a draught in a glass measure.
1872 L. P. Meredith Teeth (1878) 252 In some localities, the dentists..crowd their surgeries together in the same building.
1938 F. B. Young Dr. Bradley Remembers i. 1 Between six and eight..Dr. Bradley ‘took’ his evening surgery as usual.
1944 J. D. Carr Till Death do us Part xi. 113 I've got to be back..for surgery at half-past ten.
1964 D. Francis Nerve v. 73 I'm late for surgery... Those pills ought to keep him quiet.
1975 ‘J. Bell’ Victim ii. 23 Dr. Swallow was dealing with his morning surgery.