I haven't planned it that well, but here's the rough idea.

1. Some event happens in the 50's that leads into a war between the USA and New England. This war is primarily land gain based. USA either gains Pennsylvania and New Jersey or loses West Virginia and Maryland, Delaware, and D.C. The British Empire  is given the choice to join in the war.

2. Based off the war results, if the New Englanders lose the war, this is one point where the British can puppet the New Englanders and this leads to an integration. If NEN wins, then that's just how it is.

3. The actual civil war between the USA and New England happens after some event that is on a higher scale than the first escalation. Either the USA annexes all of New England or USA loses a ton of land. I'm really against New England annexing all of the USA because of the massive land held by the USA would be very hostile against their new overlords. This is the same with many timelines where the CSA wins the Civil War, but causes multiple wars afterwards. There aren't many timelines that outright annex them after one war decades after their inception. Something reasonable would be to have the USA be just dismantled here. i.e. Mexico gets some of its land back, The British get Oregon Country if they lost it, and the midwest in that one picture I posted above a while ago. The New English are given the choice to take Virginia, or take a large chunk of the east coast. Or for a large amount of infamy, take over the ENTIRE USA.

If the New Englanders are potentially losing the war, such as NYC/Boston occupied, the British can intervene on the New Englander side and have the situation in 2. come off