Set 20 years after the events of TNG/DS9/VOY, things are going okay but not the best they could be. The Dominion War was a wake up to the Federation. History has shown, from the Klingons and the Romulans, to the Dominion and the Borg, that sometimes diplomacy will not be enough, and force will need to be applied. Thus the formation of a new branch of Starfleet, one dedicated almost solely to war fighting. Starfleet Tactical.
It's considered a necessary evil, but one that must exist. Much like Section 31, which is indeed still active though whether or not I would ever actually use them in a show again would be in the air. Maybe if one of my writers thought of something really good, but it's doubtful.
The Romulans, though. I could potentially just rip off that garbage Kelvin timeline and say that [epic space disaster] happened and their sun started going supernova. Spock, who was visiting at the time in an ambassadorial capacity, sacrificed himself and his ship to stabilize their sun by [space technobabble here], though he only pushed back the inevitable from days to decades. (This is why he's not in the show, plus maybe I'll show edited scenes from the 2009 reboot for that exposition flashback) But given this time, the Romulans and Federation managed to stage a successful evacuation. As a plus, the Remans were given their freedom and evacuated to a new planet suitable to them because fuck them I don't want them in my show.
The Klingons remain allies of the Federation, though our relationship is rocky at times. Martok still leads the Empire, though he does grow old and there is growing talk of succession. Worf is still official Klingon\Federation ambassador.
The Ferengi reforms are going fairly well, but Ferengi society is actually not much different. The females turn out to be almost exactly like the males; lecherous, greedy, swindling bastards. And there's always a fairly large amount of Ferengi Traditionalists who cling to the old ways out of choice.
I would maybe bring back Quark and Rom for cameos, but if I had to say anything about Nog it would be that he's stationed on a starship in the Gamme or Delta quadrant and that's why we never see him now.
In a surprising twist, Cardassia is now a member of the Federation. After their devastating losses in the dominion war, at the hands of their own supposed allies no less, the Cardassians underwent a period of rebuilding and internal reforms. They'd seen what their militant ways had earned them and were resolved to change. 3 years after the war they paid Bajor war reparations partly in returning any artifacts and orbs they still possessed, and partly in vital materials and equipment though this would strain their own economy greatly as they were still in the act of rebuilding. This gesture garnered them a lot of goodwill with the Federation, who helped to alleviate as much of the strain on the Cardassian economy as possible. The eventual buildup of trust between Cardassia and the Federation, as well as Bajor's affirmative vote on the matter, would ensure their application would be accepted just 10 years after the Dominion war.
The Founders are also undergoing their own internal reforms, though occasionally a changeling will emerge from the link and choose to interact with the solids or when undertaking necessary official duties, bringing memory of their experiences back to the link as they begin to reexamine their previously held worldview.
The Jem'Hadar and Votra have both undergone more genetic tinkering at the hands of the Federation under guidance from the Founders to repair much of the damage the original tinkering has done. However, in the immediate aftermath of the Dominion war, the Jem'Hadar would fragment as a portion of them took their "Victory is Life!" shit really seriously. The Jem'Hadar do not lose. They were not designed to lose. And so we now have 2 factions of Jem'Hadar, each now being a distinctly different species as the Rebel Jem'Hadar who have defied the Founders orders to stand down are still of the same strain present during the Dominion War. For 20 years they've waged a bloody guerilla conflict against their former brethren, as well as Starfleet Tactical.
Part of the reason I would call the show New Frontiers is because as our little galaxy is being more and more explored in the Delta and Gamma Quadrants, our world is shrinking. Until someone finds some technology capable of traveling between galaxies. That could well be a major source of new shit to explore, as well as being able to explore other shit in the Delta Quadrant, Gamma Quadrant, and overall show how the galaxy is changing, as well as explore new shit in other galaxies for even more crazy scifi shit.