I think I remember the Vrill were already engaged in a conflict of their own against other spacefaring species/organizations. So I expect the Vrill intervening in form of chirurgic operations to try controlling VK sources and not really interfering with the ongoing war on Earth. After all, the bigger the outer menace, the higher the chances of a temporary armistice between the combatants to join forces against the Vrill. Although joint operations would be kind of strange as every part would be willing to keep its own tech and keep trying to steal another's in preparation of a restart on their mutual conflicts.
I for one can't see the SSU VK related forces willingly and openly collaborating with the Blutkreuz or viceversa, and I am not naive enough nor would the Allies command (or anyone else for that matter) to give access to former and maybe again enemies to both my technology, information or tactics beyond what they can deduce from battle reports or casualties. A simple study/recon by the Vrill would produce such conclusions so their policy should be one of low intensity and focused small actions. Given this policy neither Earth faction would have any interest in helping another but just defending its own resources.
So it would become more of a three sided conflict with a fourth unexpected guest, neither wished nor unwelcome. After securing all its resources, one faction can wait for a Vrill attack on another's resources to pile on and try to enact some later advantage of the hopefully more or less exhausted combatants and win an easier victory. That, if something else, could drive to more of a stalemate in the "normal" battlefront as every faction redirects funds and materiel to its elite VK powered forces and their own support assets. And maybe too to an economy of attack and a boost of navy and air forces?
An interesting scenario after all.