First Contact said there weren't many governments left after WWIII, they probably were just some City-State and not indicative of any real nation.
Also in FC we learn that within 50 years of the Vulcans making first contact in 2063, most of Earth's serious problems have been solved. I suppose that there could still be these types of kangaroo courts that Q has recreated still running in 2079 somewhere in the world, but not for much longer after that.
The fact that Cochrane is building his warp ship out of a still-operational Titan missile at an abandoned USAF silo farm, out in the open, tells me that there isn't much of a U.S. Government left to stop him. I mean, try building a large rocket in your backyard today and see how far you'll get before Uncle Sam knocks on your front door.