I think that the Nerada Incident has had a knock-on effect on all the Alpha-Quadrant races. The Federation increasing the size and power of their ships would have had a knock-on effect with the Klingons and Romulans
It would be an arms-race, and no doubt they may have attempted to keep up. IIRC the Cardassians had not suffered their economic woes during TOS, so their own designs once they became more militaristic would also be influenced by the other Alpha Quadrant powers.
With regard to the Dominion, it would depend on how much spying they did once the Bajoran wormhole had been discovered. Any exploration ships sent into the Gamma Quadrant would be bigger, badder designs, so the Dominion would probably go into a major ship-building program to compete, making their normal Jem'Hadar Attack Ships larger and more powerful, and perhaps churning out more of their huge super-dreadnought designs.
The Dominion could probably accomplish much in the years from the first Alpha Quadrant explorations up to the beginning of the Dominion War, or even put off their hardline stance for a couple more years of preparation time.