I can give you an example of a battery swap out I had. I called Apple on a Thursday. The prepaid shipping box arrived the next day, Friday. I took the MacBook to FedEx that same day. It arrived in Texas on Monday. The fixed it and handed it off to FedEx the same day. FedEx delivered to me Tuesday.
A screen swap I think would take less time than a glued in battery swap, so my example should give you an idea of what to expect.