Fallout – When will I be able to complete Underground Undercover, when waiting for Z1-14

Above half way through the main story but not by much. You can continue doing side quests even after completing the story but later in the game you're required to make some permanent choices.

The choice made at this point in the game marks the start of a quest line with each faction.

If you complete the Signal Interceptor with the Railroad, the quest you get to perform in the Institute is called Underground Undercover, in which you make contact with a scientist inside the Institute and coordinate an uprising of synths. Desdemona will be in charge and Tinker Tom will operate it.

If you complete it with the Minutemen you get a quest called Inside Job, which leads to the destruction of the Institute. If you choose one of the other factions to build the Signal Interceptor but later get kicked out of the Institute you're able to speak to Preston and continue with the Minuteman quest line. Preston will be in charge for the Minutemen and it will be Sturges who deciphers the blueprints.

If you build the Signal Interceptor with the Brotherhood of Steel you get the quest From Within. This quest tasks you with attempting to recruit an Institute scientist. Elder Maxson is in charge and the scientist is Proctor Ingram.

This choice doesn't lock you into any of these factions, that happens later in the game and is dependant on choices unrelated to this quest.