Probably the BOS due to them defeating the Enclave on the east coast. I don't know much about the west coast but I hear that they are much more powerful but I do not think that are as kind-hearted as the east coast.
Only reason why the BOS won was because of the unique technology they used, Liberty Prime and Tesla Cannon for the Vertibirds.
After Raven Rock was destroyed with Eden in it and Autumn probably killed they weren't completely organized and were easier to pick off.
It's no wonder why the BOS won that war.
Besides, the Enclave are just like the BOS with members, they don't take new recruits.
So it don't really matter which one the NCR meet.
They will always have more soldiers around the corner while Enclave and Lyons Brotherhood can't replace their lost soldiers.
With Liberty Prime dead, what do Lyons Brotherhood have that could triumph the NCR?
Nothing.
let me dig out my earlier post:
NCR
-Huge army.
-Advanced weaponry (They sold pretty awesome stuff during my fallout 2 playthrough's so I don't understand how people think their weaponry is "decent" at best)
-Unknown amount of secret weapons (One we do know of is Archimedes.(Not working right in 2281))
-Elite force known as NCR Rangers
-Do not have power armors but have pretty good armors non the less.
-Outcasts may very well align themselves with the NCR in order to take Lyons down who in their eyes is a traitor.
-Seems to have a problem with communication (New Vegas)
Eastern BOS
-Power Armors (But NCR has weapons that can penetrate their armors)
-Under-supplied. (As far as I remember they don't have any huge storage filled with stuff and they don't produce any either)
-Liberty Prime is broken.
-Still very selective when picking members. (They wouldn't pick us, the PC's, until BS, if they wouldn't pick us, then why would they pick random people?)
-Outdated weapons (Enclave using plasma weaponry while BOS using small guns weapons and laser weapons)
-May very well be under threat from V87 mutants and remnants of Enclave.
-Has no allies.
So?
How are Lyons Brotherhood going to beat them?
Just beacause they beat the Enclave does not make them the top dogs of the fallout universe.
Didn't i read somwhere that the NCR have no idea how to use Archimedes II to its proper potential? besides you cant beat Prme and his anti communist slurs!
Liberty Prime is broken.
No one knows how long it will take them to make him functional again.