There's a counter-factual you could play where the Cardassians discover the wormhole first. In practice I don't see how that happens - the wormhole aliens only allow passage to those they approve, following negotiations by Sisko.
But let's say Gul Dukat turns out to be the Emissary. The next few years probably play out in a similar way. Early contacts don't come to anything exciting. Eventually the Jem'Hadar blow up a Cardassian cruiser to serve as a warning.
Odo is still around as Dukat's sherrif, so he's equally drawn to the Omarian Nebula, thus tipping off the Cardassians to the location of the Founders' planet. The Obsidian Order-Tal Shiar episode probably plays out similarly.
Then what? Do the Klingons still invade? Do the Federation join them this time? Do the Cardassians try to ally themselves with the Dominion? Perhaps it's the Romulans instead?
Meanwhile what's going on with Bajor? Perhaps it's they who invite the Dominion to help their liberation and drive out the Cardassians, in a similar way to Dukat's motivation being defeating the Klingons. Bajor find a Dominion very responsive to them. Not only does Odo vouch for them, but they're a relatively pliant theocracy. There's a lot to work with there. Perhaps some Bajoran Vedeks come to believe that the Founders are indeed Gods, or even the Prophets themselves.
With a foothold in the Alpha Quadrant, and no mines across the wormhole, the Dominion are likely to wait until they're ready to conquer the rest of the quadrant.
