honestly, maybe on KMC they would lose, but if this was an actual fight in a comic, the Authority would take this.
lets look through this. I'm a huge Authority fanboy, so if anyone disagrees with a point I'm making, lets hammer it out reasonably...
Apocalypse
Onslaught
Mr. Sinister
Omega Red
Nimrod
Classic juggernaut
Magneto
Dark Phoenix
Apollo
Midnighter
Doctor
Jack
Jenny Q
Jenny S
Swift
Engineer
if you're giving the Carrier to the authority, Apoc, Omega Red, Juggernaut, Sinister, and Nimrod can all be tossed into the Bleed. Sinister, Apoc, and Nimrod could probably find their way back, eventually (Bleed technology is WAY advanced...) but they would have to get through the various creatures and beings that live within it...
so we're left with....
Magneto, Dark Phoenix, and Onslaught
If you've ever read the Authority, you'd realize that telepathic attacks probably won't work on either of the Jenny's, the Doctor, or the Engineer. So a quick win from Onslaught isn't going to win. Magneto would probably f--- over Jenny Sparks or the Engineer... so if the Authority were playing this smart (which they ALWAYS do) ... they would put...say... Apollo and the Midnighter after Magneto. Magneto, in the long run, would probably pull the win against them... but they would be able to match him for a while at least.
so that leaves us with Dark Phoenix and Onslaught versus Jack Hawksmoor, Jenny Quantum, Jenny Sparks, the Doctor, Swift, and the Engineer (why didn't you include Rose Tattoo... she's a member also...)
Jenny Q. and the Doctor's feats have proven that they have better matter manipulation than I have seen out of Onslaught. You can bring up Franklin Richards feats... but that would be implying that Onslaught had full control of his powers, and had F. Richards at his full potential, a claim which I would LOVE to see proof of. The Doctor has a number of weapons to use against Onslaught... time manipulation, astral plane battles, energy absportion....between Jenny Sparks distracting Onslaught into focusing on his magnetic powers on her, and a bombardment from the Doctor, I don't see him winning. It honestly would not be that difficult AT ALL for the Doctor to simply kill both Magneto and Xavier at birth... read the storyline in which the previous doctor gains his powers back from Jeroem.
Dark Phoenix is the quandry though.... although I think Jenny Q. could stalemate her long enough for the Doctor and Jenny S. to finish off Onslaught... which would allow them to help with Midnighter and Apollo versus Magneto... and then bringing the full team against Phoenix...