Hoping you can help me,
Luke van Lunenburg
Where are you at within the Institute? There are several side rooms like the Bioscience department, Advanced Systems, etc. that are contained within their own cells. Fast travel from these rooms isn't possible. It's only possible from the more open, central portion of the Institute. This may not be the case for you, but it's a thought.
One you talked to everyone and got the chip installed in your pipboy you need to return to Father. You should get a quest update to talk to him. After you talk with him and he gives you a quest to retrieve a synth, you'll see a notice appear on the top left of your screen informing you that you may now fast travel in and out of the institute.