It seems that WWIII would be likely, considering the tech and resources X-COM came across during the invasion, but there is one thing to keep in mind: X-COM was a pretty clandestine operation. While they came and rescued civilians (okay, maybe not ALL the civilians they could, with Odd as the commander
) , no one really knew who it was that was helping them. I don't think X-COM soldiers have any kind of insignia on them other than their home country's flag. Hardly anyone knew about X-COM.
Also, in regards to the soldiers, it wouldn't be too difficult to remove/neuter the more drastic effects of gene modding (like mimetic skin) or to suppress the psionic abilities in the soldiers. Realistic skin to cover mechanical limbs isn't unheard of. I'm sure X-COM would take care to tie up any loose ends that remained. They would still be ready to launch the operation at a moment's notice, but they would fade into obscurity after their job was done.
Here's what I think would happen: after the alien threat is gone, X-COM would ready their soldiers and personnel to return to normal lives. Most of them will go home, leaving a skeleton crew to stay behind and maintain/defend the base. Those members within the Council would also resume their normal careers, keeping in mind that if anyone else were to know of the existence of X-COM and the resources and tech it now possessed, it could lead to World War III. The base, and all of its secrets, would be hidden away like the ghost bases of today's military. The only difference being, they are not protected by any one country's interests, but by the interests of
every major country on earth. Everyone on earth would be thankful to those that halted the alien incursion, but they will forever wonder who it was that saved them.
P.S. in regards to MELD and weapons, you have to remember that at least 70% of our advanced technology was pieced together from stolen/recovered alien technology. Super soldiers on the level of MECs and more super weapons than X-COM released would not be possible, considering that whoever wanted them would have to get their hands on enough (and the correct) parts and someone with advanced knowledge to have the ability to make said weapon. It's just not possible if you're not X-COM itself.