I have played thru the endings with the BoS and Institute. I am playing thru with Railroad now, I am on this quest and at the point where the Synth told me to continue working with Father. I was wondering how far into the Instutue quest line do I have to go before the Synth contacts me about the rebellion?

Thanks in advance.

You need to do a few missions for Father. He will then tell you to meet with the Directorate board (he will not go). After that meeting, a synth will tell you to go meet with Z1-14.

You have to advance to the point where you get the core for their nuclear reactor. After that point you'll have an event occur and right after that Desdemona will talk to you and you'll get the ball rolling.

I have advanced thru the Mass Fusion quest and met with the board. I am now at the quest called "Pinned Down" and still no notification to progress thru the aforementioned Railroad quest.

Keep at it - you still have 1-2 more to get through.

Yep I had get thru the quest "Powering Up" and then it triggered after the board meeting that followed that quest.