The UFO universe would later become the Star Trek mirror universe

Just think about it. Stripperiffic outfits? Check. Dark themes? Check. And then there's talk of a unified Earth - so they can fight other planets better. If, say, a spaceship landed there, in 2063, and all of UFO's history with "shoot first, dissect later" had gone before, it'd be pretty much a natural reaction to shoot the newcomers (you know, before they get to use their newest secret weapon) and loot their ship.
(This is a WMG with emphasis on the "wild", just so you know...)

SHADO Assets were recycled for X-COM's use

We never hear of these mysterious invaders after their attacks in the 70's and 80's, and XCOM never faces the particular civilization responsible for the attacks. As XCOM is a multinational organization, it is highly likely that the US provided assets from their originally anti-communist unit The Bureau, and Japan provided insight, lessons, and assets from their failed solo effort in the Hiryu-kai, which never had access to interceptors powerful enought to catch a single UFO (JSDF assets are significantly less-advanced than the top-of-the-line US assets). The UK? After confirming that the invaders are not connected to their previous invaders, SHADO assets were released from their previous mission and the manpower and assets were seconded to XCOM, although due to their significant age, mostly what was provided was their doctrine and the basis for their detection system, which is mirrored in the modernized satellite system XCOM uses, and the interception/ground recovery teams are heavily influenced by old SHADO doctrine. Straker? He was allowed to retire discreetly after some talks with the Commander and Bradford, but he kind of refused to stay quiet enough, given some phone calls Bradford received during the events of Enemy Unknown/Within...