Would they really?Over in England, with the exception of Chesterton, all the other members of the collab government view themselves as the only sane people around, and that if HMMLR wins, Germany will raze England to the ground, hence all of their oppression against their own people... Yet as we can see, a Free England under neither Jellicoe nor Wilson will result in a German invasion. In fact, an victorious collab government might even make things worst, since OFN will be much less willing to send over aid in the even of Operation Sealion II.
One confusing thing about Goering's path (which is probably going to factor into the overhauling rework planned in V&J) is how the Militarists run on a platform advocating continuous conquest across the world and once they win the GCW, the OFN (and the CPS) don't take the initiative on what is clearly a huge incoming threat in meaningfully responding to Goering's wars of aggression unless a member of their alliance becomes next on the chopping block. Until then, they do little more than diplomatically coping and seething that just spams events on the Goering player's screen. That is of course a pretty illogical way to deal with an nation that is blatantly vying for world domination with little restraints when it would be obviously better to not wait for them come to strike you first and take the fight directly to the Militarists as forcefully as you can.
Consider how OTL saw the US delivered massive amounts of aid to Stalin's USSR via the lend-lease despite the massive ideological gap between the two, surely it would only be reasonable that they would go further than just undertake similar aid efforts for Collab England but also immediately throw aside any and all reservations, quickly bury the hatchet with them and put boots on the ground in England like the first time around back in the WW2. If only because of the clear danger presented by the German Reich's return to unhinged aggressive expansionism . The Collabs can at least mend fences with the OFN on the basis of historical ties, them having at least the basis of a democratic system and a shared disdain for Nazi domination. This is not taking into account the fact that all the available Collab pathways can actually join the OFN as well.