I think it would’ve looked a lot like the Dominion-controlled Gamma Quadrant. There were a few GQ planets that the DS9 crew encountered prior to the War that had been conquered by the Dominion; in fact this was how Starfleet learned about the Dominion, by encountering and hearing about their oppression.
So this is what I think would’ve happened to the AQ. No Federation, no Romulan or Klingon Empires. No large organized alliances or multi-planet governments. Individual species would’ve been siloed off from each other and prevented from organizing. The Jem’Hadar would’ve run rampant through the AQ spreading fear and enforcing Dominion oppression. Vorta administrators would’ve been installed as regional governors. I wouldn’t be surprised if the Klingons would’ve been completely obliterated from existence for their refusal to yield. Very possible that Earth and Romulus would’ve suffered the same fate as they also would’ve been sources of resistance.
I think it would’ve gotten incredibly bleak for the native AQ powers. As Sisko once said...”the Dominion wouldn’t allow anyone else’s chaotic empire to coexist next to their perfect order” (yes I paraphrased a bit). So this is how I see it playing out.
