I belong to the group of people that truly believe that the Confederacy could only have gained its independence through direct recognition and aide from Britain. (Barring of course a POD that so changes history as to make it unpredictable.)
British recognition and aide would lead to an American declaration of war. Lincoln and Seward said as much and really had no other choice.
An American declaration of war would inevitably lead to an invasion of Canada. It's the only way to take the war to Britain.
It would either be a short farcical affair that both sides rush to bring to a status quo ante-bellum ending. Or it would be a nasty brutal affair that shocks the British public and devastates world trade. Britain would have to send troops and directly assist the Confederacy. Losses would be high and Britain would probably see a change of government. Three to four years of brutal war would leave the CSA, USA, Britain, Canada and possibly France / French held Mexico bitter and battered.
In Europe Russia and Prussia would seek to take advantage of this distraction. If France is bogged down in North America Prussia may push its advantage in Schleswig-Holstien War further. Perhaps a different route to unification by brokering a deal with a nervous and isolated Austria. Russia would probably crack down extra hard on the Poles and put pressure on the Ottomans.
With its overseas trade ravaged by American commerce raiders the British economy will suffer. It will recover, but America will be wary of British investment and the CSA will prove to be a poor economic ally. Cheap Confederate cotton will hurt the Egyptian and Indian economies as British investment pours into the CSA. But there will be growing movement that sees this as "blood money" as it earns profits on the backs of the black slaves. Expect this to quickly tie into the labor movement in Britain and France as industrial workers compare themselves to the Southern slaves.
The US will bust its butt to complete the transcontinental railroad and Lincoln will be there at its completion. Though blamed by some as the man who destroyed the Union, most Northerns will be very sympathetic to him. This will increase after the disastrous term of Horatio Seymour, the last Democratic president. Lincoln will do his utmost to quell any latent secessionist feelings in the West after moving to San Francisco. In the 1870s Sam Clemens will complete his biography, a book regarded by many to be one of America's greatest works of non-fiction.
Over the next two decades Britain will tire of spending massive amounts of money defending Canada and the CSA from the US. A disagreement over escaped slaves and Central American debt payments will culminate in a crisis that nearly launches another continent wide war. In the end Britain will abandon the CSA, which will loose Texas, the trans-Mississippi, and the Upper South over the next several decades, in order to mend relations with the US.
Texas will remain independent but at odds with a Mexico damaged by the reign of its French backed Emperor. Tennessee, Arkansas, North Carolina, Louisiana, and even the old industrial heart of the Confederacy, Virginia, will rejoin the Union in the early years of the 20th Century. The Deep South will stubbornly retain slavery until the exceedingly vicious rebellions of the 1950s force the US and UK to intervene.
Alaska will still go to the US. Russia doesn't want it and they really don't want Britain to have it. The US economy will recover enough by the mid-1870s and its still a good way to geographically surround Canada.
Canada will be forced to reassess it relationship with Britain. A unpopular war allied to a slaveocracy will give the pro-Republic factions a lot of strength.
Ben