It is outright stated why they left the Federation in Season 3 and it actually was integral to the story arc, also touches heavily on the posts question.
After the dilithium supplies started to dwindle, the Federation stated pushing for alternative fuel sources or methods of faster than light travel. Ni’vars proposal for this (as it already had been unified) was a project called SB19 but they were unsure about it. They asked for permission to not go ahead with it because it was dangerous but the project was the most promising project of the day so the Federation pressed Ni’var to go ahead anyway. Then the Burn happened. Ni’Var felt that the Federation had forced them to cause the Burn so they denied the Federation any further access to SB19 data or tech and left the Federation.
This came to a head in Season 3 when Burnham was looking into the Burn and found a time discrepancy between the explosions of different ships. They had a lot of the required data to pinpoint a central point where the Burn originated from but needed SB19 to finish the puzzle. She asked Vance who then elaborated on the above. Burnham gets sent to Ni’Var where she has to argue to get the information in front of Romulans and Vulcans. President T’Rina says that SB-19 is of ‘great cultural and political sensitivity’ here. T’Rina also talks about the Qowat Milat, the Romulan sect focused on ‘absolute candor’ and says that they were vital in unifying the planet and helping both species trust one another. Also says they’re still used to help both species basically talk to one another.
Later, both of the species talk about how they agreed that SB19 was dangerous as hell and nearly destroyed the fragile peace that they had on their planet. Peer N’Raj, speaking on behalf of the Romulans, talks about how a great deal of suffering was happening on the planet due to the accepting blame for the Burn. Peer V’Kir then talks about insurgency in the Romulo-Vulcan regions of the planet so there’s a ton of strife involved there. Clearly they’re getting along but barely. N’Raj keeps repeatedly talking about the emotional cost and considering he speaks on behalf of the Romulans it goes to show that they are not happy about this situation. Sort of despondent about it. Their lifespans are also relatively long so this SB19 stuff is within living memory for them and not very far flung. It’s still a sore and open wound. The Romulans even threaten to go behind the back of the Vulcans and give data to the Federation. Burnham then defuses the situation by bringing up Spocks attempt at reunification and not wanting to see it destroyed over this request so she withdraws it.
I don’t know how they re-unified but honestly? I don’t think they really did. I think that it’s a process and a step and that the episode Unification III in Season 3 of Discovery was named that to demonstrate it. Ni’var is just another step in that reunification but it isn’t reunification in and of itself. It’s a long road they’re going to be traveling for a very long time.