The length of time it takes for a cell tower to become fully operational can vary greatly, depending on who owns the tower, what space contracts are in place for it, the lead time needed for technicians and equipment to arrive on site, etc. The cell companies themselves actually own relatively few towers, with most of them owned by specialty companies that can lease space to multiple carriers, and even some non-cell services.
You can check who owns towers in your area at
Cell Phone Tower Search. Note that not all towers are listed though.