In the movie Star Trek: First Contact, future Borg tried to contact their home world in 2063. If the communication was successful and the Borg from the Delta Quadrant came to Earth in 21st century, there'd be two Borg queens.
The Borg queens had individuality which represented the collective. And, technically all Borg were able to connect to both queens. How would the collective work then?
We actually have several examples of disconnected groups based on the same technology:
All of these were based in the same Borg technology, but the main Collective was unable to influence/break/fully reintegrate them without special assistance (and with the sub-collective, it didn't work at all).
In Unimatrix Zero, Part II we also learned that the hive mind allowed for instantaneous galaxy-wide transmission. Axum's ship was on the other side of the galaxy, yet there was no lag when he conversed with Seven.
This results in 2 main possibilities I can think of:
If they were already in contact with each other, I suspect the two Queen minds would "fuse", sort of creating their own sub-collective. It could act rather similarly to the sub-collective from Survival Instinct - they were part of the whole, but could still separate themselves as a group from the whole.
And if they were not in contact with each other? In VOY 5x02, Drone, the Doctor's mobile emitter is accidentally combined with Borg technology to create a single Borg more advanced than the entire rest of the collective - and the Borg try to assimilate it.
Therefore, if two incompatible Borg Collectives existed, I would expect each one to try and assimilate the other. The more advanced Collective to gain more drones, and the less advanced one to make itself more advanced.
I believe that had the future Queen's plan in First Contact succeed, the previously existing queen would have abdicated to the future one prior to merging the two Collectives (future drones with past drones). This is of course assuming that there wouldn't have been a way to merge the consciousnesses of them (if there was, then this question would be moot).
Reasoning:
The future queen would have been "closer to perfection" than her present day counter part, having had the knowledge of more assimilated species and consequently being more technologically advanced. As the Borg's quest is to achieve perfection, it wouldn't make sense for an outdated queen to hang around.
And on the outside chance that the queens would be unwilling or able to reach a compromise and came to using conflict to settle the succession, and if the past queen won that conflict (most likely outcome due to superior numbers), then all the knowledge and experience of the future queen would have been instantly passed on to the past one upon assimilation.
Season one of Star Trek Picard establishes that each cube has the capability as serving as a Queen's headquarters, and that any drone could serve in the role. Typically a pre-selected one--as the Queen indicated Seven had been identified as a potential Queen on Voyager--but given that Hugh fully expected to be able to fill the role on Picard, probably any drone could do it in a pinch: it's just that any random drone wouldn't have the self-volition to take the initiative.
This being the case, that the Borg had prepared for the possibility of having to elevate a drone to Queen on an emergency basis, suggests they also considered the possibility of multiple Queens caused by, say, a large-scale network failure that isolated sections of the Collective and how to reintegrate them back in. If that were true, then they'd have some kind of procedure to handle it. The First Contact situation would simply be a variation in which the separation between the main Collective and the isolated sub-Collective was temporal rather than spatial, and they'd handle it however they would handle the more mundane version.
