It’s hard to say how long the wait would be as it can vary widely between speciality and location. I would search Rheumatologists in the location you will be in and email them explaining your condition and medication requirements and enquire about how long it takes to get an appointment. As you can imagine, city areas normally have shorter waiting lists than regional areas.
Pharmacies wouldn’t stock Humira as an on hand drug due to the cost. If a particular pharmacy has a number of clients on the drug then they will most likely have supplies on hand. However there is no reason it couldn’t be obtained by the next day in most areas or even the same day in the city if ordered before about 11am that day. Bear in mind that if you are going to more remote areas that are major distance from urban areas then the that could pan out to 2 or more days. I live in an inland rural area between Sydney and Brisbane, so about 600kms from both cities, and the pharmacy is able to order one day and receive the next.
You won’t be able to get around having the original script on hand to present to the pharmacist, so unfortunately snail mail will be your only option. Once you have that original script then you can have your initial fill and refills done anywhere.