They planned to reunify, but during the five long years of separation Herschel Supremacy formed their own quasi-national identity. HS would be more authoritarian and militarized than the GTVA, with Supreme Admiral Peters holding most executive powers and society mobilizing by fear of both Nightmare and Shivan threat. It would also be far less corrupted and better organized, as Peters carried out lots of reforms and successfully cleared the Wolf's Group of Cerberus' influence. It would not be a bloody dictatorship, rather something like Martian Congressional Republic in Expanse, with its society proud, disciplined and eager to serve their motherland.
When contact with the GTVA is re-established, the Alliance would be in the middle of Shivan invasion, so both factions must closely cooperate putting their differences aside. But still GTVA would be very unhappy to see their former colonies so independent, united and having their own agenda. Further in the campaign they would also have different view on the Nightmares - Sinclair has convinced Peters to cooperate with them while most GTVA officials still see them as no less threat than the Shivans. So there would be tensions between both factions, close encounters, provocations, perhaps some special operations, but Shivan threat (and our heroes' actions) would prevent them from escalating into an open war. It's possible that war between GTVA and HS would broke out after the Shivans are defeated, but it was never planned to be in the campaign.
