My main issue with the post American Civil War Turtledove books is that I am very sceptical of the United States itself surviving a situation where it attempts to use military force to get the seceding states to return and fails for some reason. This is partly due to the lack of prestige and the precedent that state secession will work. Its partly due that whatever causes the failure would be a radical enough POD as to shatter the United States.
The South getting modern weaponry would actually be the one situation that would work with the USA surviving the defeat since being defeated by weapons sent from the future would be a good enough excuse for the federal government to keep some prestige, plus the "Guns of the South" occurs at a point where the federal forces had started to win some victories. Another situation that might have worked is the federal government and the other thirty some states just letting the six Deep South states go peacefully. But I think anything else winds up shattering the USA. And if not, the North just keeps trying again and again until eventually they reconquer the South.
Also agree that we are probably going to get World War I butteflied away or into something entirely different in this world.
But yes, there is a good chance that the CSA is aligned with the British empire due to lots of economic (its their main trade partner) and strategic reasons. There is also a good chance that Woodrow Wilson is the Confederate president during this period.