Did some !!science!! with another fort.
1. Created two hospital zones, one containing beds, the other chests of supplies.
2. Forbid all other cloth, thread, soap, crutches, splints, etc. in the fort so they have no choice but to use the hospital stores.
3. Dropped some soldiers 5z to give them some moderate injuries.
4. Wait for them to go to hospital beds.
5. Observe as doctors happily fetched supplies from the chest zone and brought them to the bed zone.
Conclusion: Dwarves are perfectly fine with performing medical labours across multiple hospital zones.
I tested it out too and i an confirm, it's working! I made 3 zones as hospital, one with storage, one with beds/tablets/traction benches and one just for 3 nice, fresh water wells. Then some accident with naked goblins in cages happend in my dinning hall happend... 5 dwarves land in hostpital, They was checked and taken care fast. Dwarves are looking like They love only wells in hospitals, if there is one in zone, even if closer are drinking zone wells.