How can I progress with the Brotherhood of Steel quest line after "The Lost Patrol"?
I have completed The Lost Patrol mission and it is in the completed part of my journal. I also completed the side missions (The radiant ones you get to collect tech for Rhys and Haylen) but nothing happens after.
I am sure the next one is where you go to the airship but nothing is happening. Nothing is even showing up in my journal, all I have are the radiant quests.
How can I proceed with the Brotherhood of Steel quests?
Complete the main story up to (and including) the quest 'Reunions'.
The Brotherhood of Steel's quest line doesn't really progress until this point. The Prydwen (The Brotherhood Airship) will show up right after the main quest Reunions is finished, laying anchor over:
The Boston Airport
After this, the Sole Survivor can visit it during the Brotherhood of Steel quest Shadow of Steel offered by Paladin Danse.
Without knowing how far into the main story you have progressed, it's hard to give specific guidance as to how to reach it quickly, but if you haven't gone to Diamond City yet, that would be the best place to head to continue the main story quest.