New England doesn’t even have to fight a war because the UK would recognize them immediately and maybe even offer military support if US tried to invade. New England is right next to Canada and has heavy enough ties with Britain to do that. The US accepts succession of New England but agrees any additional states or regions that try to join New England won’t be accepted by either side and borders are recognized as they currently are(upper New York is like Missouri in this situation but New England and Britain agrees with the union not accept any part of that to lessen tension since trade and businesses will continue as before but New England has more control over its affairs).
In the wildly unlikely event that New England attempts to secede, there will be a war. That's the way it always went in the 19th century and the first half of the 20th century If Britain recognizes the New England Confederation, that just means it is a bigger war. This wouldn't be like the Trent Incident, the Union would be a lot more powerful. Looking at the
Rise and Fall of the Great Powers by Paul Kennedy, by 1880 the US had almost double the manufacturing of France, Germany, or Russia; and matched Britain sometime around 1890. Obviously, the US wouldn't be doing that well if they lost New England, but the presence of the hostile, expansionistic Confederacy to the south means the US would have a significantly larger peacetime army and navy than in OTL. Britain's ability to project force into North America is more limited than the Union's. With the amount of rail connections between the US east and west coast, there is no way the Union could be blockaded, and Britain has far too powerful a fleet for a blockade of New England and Canada to work. At sea, I'd expect rival commerce raiders to cripple the merchant fleets of both sides. The Union would probably try to conquer Canada again, with a decent chance of getting everything west of Ontario and probably regaining some of New England. Britain would probably keep its most populous Canadian provinces and most of New England, so both sides would declare that they won the war, when the real winners would be France and Germany. The New England Confederation would probably find itself a British client state, with no more, and possibly significantly less control of its affairs than before secession.
In the middle of this mess, Confederate hotheads could lead an attack on the Union, intending to "liberate" Missouri, Kentucky, and Maryland. I'd expect this to fail, with the Confederacy losing Tennessee and Arkansas, and the Union remembering this stab in the back for a long, long time.
This changes things even more greatly. Now down south during this same time you could have the Confederacy lose Texas but retain the rest of its country and maybe the Indian territory becomes its own country too(likely a puppet of someone after independence).
The entire TransMississippi could end up seeing the Richmond government as distant, unresponsive, and only caring about getting taxes from the TransMississippi, so the Confederacy might just lose everything west of the Mississippi River. The Confederacy would go to war try to prevent this, but their force projection capabilities were poor. I'd expect the Union public to favor recognition of the TransMississippi, giving the Confederates a dose of their own medicine. The Union governemt might formally recognize the New Republic, calculating that the Confederacy would be extremely stupid to attack the Union while fighting it's own Civil War. I'd expect the Confederate government to be that stupid and to lose. Though technically independent, I'd expect the TransMississiipi to become an economic client state of the Union.