The Railroad ending and the Institute ending in Fallout 4 lead to the destruction of the Prydwen and the Brotherhood of Steel, respectively, ruling out those options. If the Brotherhood ending is canon, the Sole Survivor becomes a Sentinel in the Brotherhood.
Clues from a show set nine years after Fallout 4 suggest that the Brotherhood of Steel relocates the Prydwen to California after losing control of the Commonwealth to the Minutemen. This move may have been necessary to support the West Coast Brotherhood, which is weakened from a war with the NCR.
The Sole Survivor, either Nate or Nora, may play a significant role in the Brotherhood of Steel or continue helping the Minutemen in the Commonwealth. The latter scenario is considered more likely for the TV show’s setting on the West Coast.