Hi MBM and welcome to the Lincoln Forums. Action did a great job of covering your question and its numerous variables.
In normal, pre-pandemic times, one to two weeks would usually be an average time line for an appointment with a loaner. But these are not normal times, so scheduling timelines are extended. And it will vary form Dealer to Dealer. Some Dealers are not even able to currently offer loaners at all.
You can also call Lincoln Concierge to see if the can help. In the case of Dealers with no avaialbe loaners, they currently offer up to $45 a day towards a rental car.
Also, you don't mention what type of warranty service you are having performed. One day in the shop, multi days etc.? If you can live without the loaner, you would probably be able to be scheduled sooner. In cases like that, you can use the Drop-off/Pick-up service (The Dealer will come pick up your vehicle and then drop it off when it is finished).
Or if you wish to drop the car off yourself, most Dealers also offer rides back home, then will pick you up when the vehicle service is finished.
Mine usuall (pre-pandemic) had "runners" who will bring you home/pick you up (usually retired men/women they hire for the job). Now, they arrange Uber rides, which they pay for. My guess is for liability reasons right now.
I would recommend calling Lincoln Concierge to see what other options you may have.
Hope some of this may help, let us know how you make out and good luck.