Well, the story makes it hard to mesh with the small-scale gameplay, with the dramatic and public assault of Mumbai and everything. Story is great and the Mumbai thing is important for the theme of the game, but then you wonder why these big superpowers are not doing much and your squad of 8 guys is doing so well.
So after these bold assaults, it unfolds into a slower, less wide-scale conspiratorial shadow war between the aliens and PHALANX. Like spy vs spy, or terrorists vs police scale conflict, more subversion and skirmishes, not so much world war III.
There could be some background story to explain it, say the aliens attacked just to test the waters, or it was just an extremely violent science expedition and decided open warfare would be too costly. Maybe there are not enough of them to sustain full conflict. Or they realize Earth's governments will use nuclear weapons to render the planet sterile before they ever lose an invasion. Anyways, they are 'alien' and have unknown technology and intentions, they don't need to rationalize like people do. Maybe the violent incursions worked historically for them to cow the world into surrender most of the time and decided to try that out first?
I think nuclear war would be a good losing condition if 5 nations defect and there are no other nations above 'pleased' or somesuch. The remaining governments decide defense of earth is a lost cause and they press the red button, destroying everything else the aliens would possibly want to take, just to spite them, but losing the game for the player.
As for the non-PHALANX militaries, I'd suggest and I've always imagined that they only act if they have to. Say you shoot down a UFO and leave the crash site too long. In XCOM, the crash site would quietly go away with no complaint. Maybe the host nation was just grateful you shot it down, they blasted the site with artillery or bombing runs and salvaged what they could, either way, not your problem. I imagine during terror sites, they hold a cordon to keep it contained, but are not competent or willing enough to go in an fix it, unless you procrastinate for too long.
As for like, cops or soldiers running around on a map battling aliens, they'd be little more then red-shirts anyways, so why bother coding them in? Either way, I think the world's military would be better represented with a little message saying they assaulted and recovered that crashed UFO you never got around to, or random crash sites appear in places you don't have radar coverage of. They managed to shoot one down and they'd appreciate it if your super-soldiers would come deal with it.
I think in the current version, shooting down a UFO over land and ignoring the site is like ignoring a harvest or intentional landing, which I think is bad. In the old Xcom you got points for shooting down UFOs, even if you just left them alone afterwards, you just got more points for attacking the crash site. I think countries should OK with crashed UFOs on their land and just give you credit for shooting them down. Or they shoot them down and you attack the crash site. They give you half credit for whatever half-assed work you are doing.