Well, the All our Yesterdays sourcebook has some ideas for this very scenario. Basically, the Feds are getting pasted, and even the advent of new ships like the Sovereign and the Defiant aren't helping too much. Predictions still say the Federation will fall until the Dominion make their appearance and the Klingons begin to consider allying with Starfleet against the threat.
A 20-year war might not be such a reach. Remember, the Federation Alliance weren't seen to be capturing many planets during the Dominion War - they went straight for the jugular at Cardassia Prime, and had cut the Dominion off from its primary supply line to the Gamma Quadrant. Given that the Klingons have all their resources in easy reach, the Feds in the alternate history were "complacent" having never been "kicked" by the Borg and the Romulans may have been covertly supporting the likely winner, I'd say a twenty-year war could happen. The Betreka Nebula incident lasted 18 years, after all...
"That might have been the biggest mistake of my life..."
"It is unlikely. I predict there is scope for even greater mistakes in the future given your obvious talent for them."
Vila and Orac, Blake's Seven