Every alternate history story I have ever read about the US Civil War in which Great Britain intervened, resulted in a CSA victory. I am not sure if I buy that. Recognition of the CSA and military intervention by Great Britain solves some of the South's problems, but certainly not all of them. First I question what level of military commitment Great Britain will provide. Certainly they will break the blockade which solves some of the South's economic problems as well as helps ensure that the Confederate Armies will be properly supplied. However it does not give the South adequate road and rail transportation, it does not give them a larger field army, just a better clothed, equipped and fed one. It does not solve the problem of a lack of national unity that the USA had. Great Britain would supply adequate naval force, but any troops sent to North America would most certainly go to Canada to protect it from US invasion, not be sent to defend Richmond. As for Canada, the United States has the advantage there, the US would certainly control the Great Lakes and the St. Lawrence waterway could be easily blocked. The US West Coast was vulnerable, but any losses there, while painful would not be catastrophic. Lastly the loss of corn and wheat that Great Britain imported from the US would certainly be felt, plenty of cheap cotton is great, but you can't feed your citizens with it.
The question is, could the US defeat both the CSA and Great Britain and still restore the Union?