It varies by location and their policies, but usually the inmate will be able to set up their telephone list within 1-2 weeks if your are immediate family, a little longer if not. Letters may be sent as soon as the inmate has money on the commissary to buy stamps.
https://www.inmateaid.com/ask-the-inmate/how-long-does-it-take-a-prisoner-to-communicate-with-their-family-once-they-arrive-to-the-correctional-institution#answer
Accepted Answer Date Created: January 29,2014