Canonical Ending Predictions
Winner: Railroad/Minutemen alliance/Evacuated Institute
Institute: The evacuated Institute spreads to other states and abandons the Commonwealth, taking over former science-installations and facilities which they fill with Synths and Coursers. Future Vault Dwellers will be do missions for the Institute to recover scientific technology or mad science experiments ala the Big Empty. They will serve as the Followers of the Apocalypse/Mad Scientists of future games.
Brotherhood of Steel: Having lost Maxson, Prydwen, and Liberty Prime, they will proceed to be far weaker than before but still strong. Isolated patrols and chapters like the ones in Cambridge will be encountered as well as people trying to actively recruit new members. They won't be a major Empire from now on but just one faction amongst many.
The Railroad: With the Institute destroyed, they will switch over to being a generic anti-slavery organization and help you smuggle various groups like ghouls, Synths, and regular slaves away from the people who want to be wiped out.
The Minutemen: Will become the rulers of the Commonwealth and will hire the PCs as bounty-hunters and/or work to spread the law of the Commonwealth Republic. They'll be the beginnings of a much-weaker NCR group.