Skyrim would be the last to fall. Take Morrowind, Cyrodil, Valenwood, Elwesyr, Hammerfell, Hightrock. Then attack from all sides. And remember, the forces of you're defeated enemy could be forced to fight, easy enough to overwhelm the nords then send in the heavy duty KPT.
As for the river Niben. Its not just a supply route for the KPT, its a supply route for the Imperials, especially is they are cut off from all angles. Just build a dam, dry the river out. Let them die of thirst.
As for black marsh. Could just take alot of sand from the deserts and deposit it into the swamp land, cut down the vegetation etc. I wouldnt instantly send my forces into Black marsh. It would be suicide.
But who says you have to invade every promise? I'd probably try to recruit some (most likely the redguards) against the empire. Promise them independance away from the empire, or help them rebuild Yokuda and what not.
Damming the river will not kill them.
I'm sure there are plenty of water sources around. And it's also not the only supply route into the province, though land transport would certainly slow things. But if they blockade the KPT from brining supplies that way then Morrowind and all the other provinces could still send supplies to the bay, where they could be carried by land. :shrug:
I'm sure that blocking off the river would not end up killing many people due to thirst though. It's more important for supplies, especially the KPT then anything else.
If there was enough of the KPT, which, since Anvil wouldnt be TOO hard of a target,
They could ally with Redgaurds, send a force with them into High Rock, and take Anvil with the help of the Tang Mo.
But the fact that they would have to sail around an entire continent leaves many problems.
One is, no matter what your are, being on a ship that long with that many people, your soldiers aren't gonna be happy. Plus sicknesses spread pretty fast in such tight areas, so thats always a possibility.
Then there is the possibility that your gonna get spotted by unfriendly eyes, unless you take some super-wide route to Anvil, which only leads to problem one being made worse.
Then think how the Redguards or Altmer are gonna feel when a strange fleet enters their waters. They don't know what intentions the KPT have, though if they recognize them as the KPT, then I'm sure they won't see them as allies at first. Plus they have to sail past either the Nords to the north or the Khajiit and Argonains to the south (though I have no idea what kind of navy either of those races has).
Hammerfell may not be on the best of terms with the Empire, but why would you trust some unknown group of leaders who offer promises of freedom? No smart leader would accept it right away, because it has trickery all over it.
The KPT would need to show their intentions with gifts and not attacking Hammerfell, plus the Redguards would need to be absolutely sure the KPT are going to win before joining up with them because should the KPT be driven off, then they are in a sore position indeed. So thats even more time the Redguards would remain probably neutral as they waited and watched who was doing better in the war.