There was a status quo within TNG, a finely balanced alignment of powers which one day could have even led to galactic peace with people like Spock championing the importance of unity between species.
The destruction of Romulus changes everything, much in the same way as the
destruction of the Borg Collective would have done so in the Delta Quadrant. All that we really know, is that the destruction of the Romulan sun caused a *massive* refuge crisis, with the Federation having to take in Romulan citizens. All of this eventually leads to the attack on Utopia Planetia and the subsequent paranoia in regards to synthetics and AI which adds to the dystopian feel of things.
The Alpha Quadrant had already suffered *massively* in the aftermath of the Dominion War, the Cardassians themselves almost being wiped out by the Founders, I am not sure what the final Cardassian losses were, but they would have been substantial to the point of being society changing, with it being highly likely that the Detapa council would have taken over from the High Command… there was likely a high level of political instability within Cardassian borders for *many* years after the Dominion War.
25th century Nu-Trek has not shown us much of the new Galactic ‘order’ of things, with the stories being quite ‘ship in a bottle’ like, keeping intergalactic politics out of it, or at least keeping them vague. I can only speculate that the destruction of the Romulan sun would cause massive upheaval and disorder within the Beta Quadrant that other species such as the Klingons *would* have taken advantage of, as the centuries old status quo would be no more. The finely balanced scales of peace would have been irreversibly tipped to the point where other species, perhaps even the Hirogen, would have taken advantage of the unbalanced order of things. Romulan space would make for a great Hirogen hunting range…
I do not think that the Ferengi Trade Union would have interfered *too* much with what is left of the Romulan Empire due to their lucrative new trade treaties with the Federation, as negotiated by Grand Nagus Rom.