It depends on how "Urgent" it is to get that site operational! They can work super fast, or at a more casual rate. Yes 3 weeks seems like a good estimate once the tower is up. They might bring in a sea container size trailer, set it down Monday and have it working on Tuesday! In that case, everything is pre-assembled, with only a 200 amp electrical connection and antenna wires needing to be installed after delivery to the site.
I worked in several buildings where space was leased to cell sites. Basically if I opened the equipment room and it was air conditioned, then the electrical meter room was turned into a cell tower radio room, and the ladder to the roof was in that room too. Radio's on the walls, and antenna's on the roof.
Many cell sites in the city, they want the tower to be less than 50' off the ground, and spaced every mile or two. They do NOT want any cell site to pick up all the cars and people 4 miles around the site, but limit it to only about 2 miles, so that the site is not picking up more than about 1,000 cells at any given time. In rural areas, all those rules are changed, so that the site can handle 500 - 1,000 customers who will be 3-5 miles from the antenna. Some can even reach out 8-10 miles, say if they are located near a National Park, and that area is 'protected' from having a cell site located near it. They might even put a site at the top of a hill.
Good luck!
Fred.