The timing depends on where the case is in the legal process, and there are several distinct phases that each affect where your person is physically housed.
If the federal charge has just been filed and the case has not yet been adjudicated, your person will likely remain at their current facility until the federal court proceedings require their presence. At that point they will be transferred, typically to a county jail in the jurisdiction where the federal indictment was handed down. That transfer is arranged by the US Marshals Service, who are responsible for transporting federal defendants to and from court appearances. The timing of that transfer depends on when hearings are scheduled and how quickly the Marshals can arrange transport.
During the court proceedings, whether that involves arraignment, pretrial hearings, trial, or sentencing, they may move back and forth between their current facility and the county jail near the federal courthouse. This can involve multiple moves over an extended period depending on how long the case takes to resolve.
Once the federal case is fully adjudicated and a sentence is imposed, the Bureau of Prisons takes over and designates a federal facility. That designation process and the subsequent transfer to a permanent federal placement can take weeks to months depending on bed space and logistics.
One important piece of good news in this situation is that time served counts. Every day your person spends in custody during the federal proceedings, whether at their original facility, a county jail, or in transit, counts toward their accrued time and will be credited when the federal sentence is calculated.
Throughout all of this, mail is the most reliable way to stay connected since phone access and accounts may not transfer seamlessly between locations.