About one week, and the timing is tied to how the BOP structures its orientation process. The Bureau of Prisons runs Admissions and Orientation on Wednesdays at most facilities. That is the formal day when new inmates go through the full intake briefing, get their accounts set up, and have phone and visitation access activated.
The exact wait depends on what day your person arrived. Someone who comes in on a Tuesday only has to wait one day until Wednesday orientation. Someone who arrives on a Thursday is looking at nearly a week before the next A&O session rolls around. Either way, once Wednesday orientation is complete, phone and communication privileges typically open up within a day or two as accounts are processed.
Send a letter with your phone number the moment you know they have arrived. Mail moves through even during the orientation window, so they can have your number in hand and ready to use as soon as their phone access is activated.