New Orsinium pretty much relied on the endorsemant of the Empire to exist without everyone else trying to wipe them out of existence, due to the infamy of the previous city-state to hold that name.

So is there any word so far on how New Orsinium is faring with the crumbling of the Septim Empire?

I could see the Redguards holding another genocidal attack on them. The Bretons would likely join in on the fun, if it benefits them.

Otherwise, absolutely nothing has been said about Orsinium.

Orsinium is allied with Queen Elysana of Wayrest. That's not a pairing you'd want to mess with, Empire or no Empire.

Besides, I'm not sure the Empire was what was keeping Orsinium safe. After the Warp, Uriel was basically forced to acknowledge a situation that already existed--Orsinium had become a legitimate High Rock kingdom.