British and French "Recognition" of the CSA...
...doesn't , necessarily, mean an alliance.
First, a lot will depend upon the nature of the battlefield situation in, let us say, the summer of 1863.
One can imagine, after a Confederate victory at Gettysburg, that France and Britain recognize the CSA - but it doesn't necessarily follow that either of them will offer more than material aid. It's quite possible that while Britain and France might no longer "respect" the Union blockade, they still might be loath to use their fleets to disperse the Union fleet.
While the Union might fear a two-front land war (i.e. Canada) - it's still by no means clear that the Brits would want to deploy troops. An invasion from the North might well galvanize a faltering Union as much as encourage a quick peace. And in any event, what is the advantage to open warfare for the Brits?
If, the Brits, can divide their fastest growing industrial competitor, the USA, simply by diplomatic recognition, blockade breaking, and an offer to "broker" a peace...Why then must it be a shooting war?
Similarly, other than following Britain's lead, what does France gain by recognizing the CSA? Would French "interests" in Mexico, be any less threatened by a victorious CSA as opposed to OTL's victorious Union?
I think that it would be reasonable to suppose that the CSA's (and Texas') ambitions with respect to Mexico would, after a CSA victory, quickly come into play.
And the need for any "alliance" (or even diplomatic recognition) becomes even less clear if Lee not only wins at Gettysburg, but goes on the capture Washington, and maybe Baltimore too - and even goes on to threaten (perhaps) Philadelphia (a la Ward Moore's "Bring the Jubilee").
The fact is that the CSA's fate, as Lee among many others saw, was ALWAYS dependent on it's prowess on the battlefield - and the extent to which the Union was committed to the fight.
As Shelby Foote put it in Ken Burns' "The Civil War" when asked "Did the South ever have a chance of winning?":
"I think that the North fought that war with one hand behind it's back. At the same time the war was going on...In the spring of 1864, the Harvard-Yale boat races were going on and not a man on either crew volunteered for either the Army or the Navy. They didn't need them. I think if there had been more southern victories, and a lot more, the North would simply have brought that other arm out from behind its back."
Look, one of the pleasures of this forum is to invent scenarios that fit our fancies. And the notion of the CSA, Britain, and France all allied against the Union can be seductive. But if we're going to make it so, we have to really consider how - and not depend on a Turtledove-ex-machina.
Still, it sure would be interesting (and taking - maybe - half-a-page from Turtledove's "Guns of the South") to imagine a Union which conquers Canada while the CSA takes over Mexico. (Perhaps then the Union would "re-invent" itself after such a conquest - sort of the way Britain re-invented itself and it's empire after losing it's 13 North American colonies.)
And if that all happens, then the Brits might well ally with the CSA - while the French ally with the Union.
If so, then, who do Prussia and Russia ally with? Does Spain "sell" Cuba and Puerto Rico to the CSA?
Even so, does Britain seek another war with the Union to regain Canada? (Why, for vengence...?) And, even with an CSA-GB alliance, what is the CSA's motivation to participate in such a war? (For OUR entertainment...?)
And, other than tweaking GB, what is France's motivation to "ally" with the Union? If they've been preempted by the CSA in Mexico (and the Caribbean) - what do they have to gain? Would an alliance with the Union realistically provide a counter-balance to the rise of Prussia? And, even if the French felt it so, what would the Union stand to gain?
For that matter, one of the cliches of ACW alternate history, is that if the CSA had gained nationhood, that long-term and continued enmity between the Federals and the Confederates would necessarily continue.
While I'm sure that they wouldn't necessarily go on to be as friendly as the US and Canada have been in the 20th/21st Century... And, certainly there would likely be border issues, and (maybe) bigger questions about Missouri, Kentucky, Maryland, Washington DC, and (particularly) West Virginia.
While it's easy enough to imagine all this leading to a new war - it's also just as possible that the border issues wouldn't necessarily have to be the reason for a new war.
Heck, with the slavery and states right's questions settled (as it were) by Southron independence, it's quite possible to imagine a US-xCanada and a CSA-xMexico-xCuba peacefully coexisting - and both nations trying really hard not to get involved in any European adventures. Probably they would each be the others number one trading partner.
I know, too much reality spoils the game.