Fallout – End game patrols

I have limited experience since I'm only on my second playthrough however so far I have observed -

If you join the Minutemen you'll end up allied with the Railroad and will end up destroying the Brotherhood of Steel and the Institute.

Additionally, if you've progressed through the Minutemen missions (upto and including retaking the castle) then during the mission where the Institute attempt to "invite" Wallace to join their Biosciences team it will be the Minutemen that respond to Wallace's distress call (thus giving you the opportunity to talk them out of it). If you haven't joined the Minutemen, the Gunners will turn up to respond to Wallace's call for help. Progressing through the Minutemen story will also result in numerous groups of wandering Minutemen patrolling the wastelands in addition to unlocking artillery.

If you join the Brotherhood of Steel, you'll declare war against the Institute and Railroad while being allied with the Minutemen.

If you join the Railroad you'll be allied with the Minutemen and destroy the Brotherhood of Steel and the Institute.

If you join the Institute you'll be allied with the Minutemen and destroy the Railroad and the Brotherhood of Steel.

After completing the storyline allied with the Institute, groups of synths start showing up all over the commonwealth announcing that the Institute is now in charge - they'll also set up a variety of checkpoints and roam the wastelands killing mutants and raiders. The various companions I used during my playthrough didn't mind that I had sided with the Institute however I expect that Deacon and Danse might (since you destroyed their factions).

Regardless of which choice you make,

"Father" will die, however the cutscene will be different if you aren't allied with the Institute. If you attack the Institute you'll lose that as a fast travel location. Any factions that you wipe out will not be available for missions after the final cutscene.

The Battle of Bunker Hill is a quest that is given to you by the Institute, which is why they weren't interested in being warned - it was their mission.

The reason you're allowed to inform other sides is because you'll always get this mission regardless of faction allegiance however if you're working for the Railroad they'll want informing because the synths you've been sent to kill by the SRB are in the process of being "saved"/smuggled out of the Commonwealth by the Railroad.

If you're aligned with the Brotherhood of Steel and warn them, then they'll request that you kill the synths.

At this point in the game it is possible to still be on friendly terms with every faction, if this is the case you can turn up to this mission and walk through multiple opposing factions trying to kill each other, with all of them acknowledging you as on their side.