Part of my current AH time line (besides the on-hold Great War) features a more common-sense approach than Turtledove to the concept of a US vs. CS + UK war. The POD isn't the same, but I'd like some input as to what sort of grand strategy you could see the powers involved coming up with.
The year is 1894, and while a general European war is currently being waged between [redacted] (no hints!

), the United States and United Kingdom come to blows over a rather unimportant diplomatic incident that spirals out of control very rapidly. While commercial ties between the countries are almost as important as in our world, moderate American anger about a British intervention during the *Civil War and the presence of a thoroughly populist, anti-big business government in the more important two of the nation's three capitals has brought upon war. The Confederate States of America have been presented with the traditionally unreasonable demands given by a power that is itching for a fight with another, weaker one and its government, two years old and barely out of the hands of the rich white planters which dominated its politics up to that point, must reluctantly prepare for a rematch almost 40 years in the making.
As in any divided-US time line, it's impossible to start a discussion without looking at the national borders. The earlier Civil War, and the British intervention, have given the Confederacy what I consider to be just about their 99th percentile of territory, industry, and population given a traditional 1840-1870-era Civil War. The (rough) map at the bottom of this post shows a Confederacy that spans from the Columbia River and the Pacific Ocean in the Northwest and West, stretches deep into the Great Plains in the Midwest, and rests its Eastern flank on the Mason-Dixon line. Roughly speaking, the populations of the US and CS in this time line are 45 million and 30 million respectively. Canada's population is similar to that of OTL at around 8 million.
The United Kingdom is engaged in a maritime war that is concentrated in the Baltic, Atlantic, and Mediterranean theaters, and has several small groups of soldiers undertaking various military expeditions in Europe and elsewhere across the globe. Few additional troops can be spared for Canada; with the exception of the small British forces regularly stationed in Canada, the Canadians must fight the US alone, though perhaps some Australian and New Zealand contingents could be transported to British Columbia.
The United States does not have a particularly strong regular army, though it is undoubtedly larger than the tiny force it was at this time in our world, at perhaps around one or two hundred thousand men. Its real strength however, is its reserves of manpower. Assuming that the United States can eventually (and not all at once!) field a percentage of its men equal to that fielded by Italy in 1914, eventually some 6 million troops will be able to march to the front (in a 3 year war, presumably fielding roughly 2,000,000 at any given time after the first call to arms).
The Confederacy is likely to resist with every resource possible. Surrounded diplomatically with the presence of small numbers of US troops in Mexico, and with the US annexation of Cuba in the 1880's, the nation appears to have been surrounded by its former countrymen. Assuming that the Confederacy, excluding its large black population, can teeter on societal collapse as in OTL's Civil War by sending a proportion of its men to the front roughly similar to that of France in OTL's Great War, we can see that the CSA can eventually mobilize some 4 million troops, though certainly not all at once (perhaps 1,000,000 to 1,500,000 at first mobilization, and roughly the same at any given time).
Take a look at the map, and feel free to ask questions or make (supported!) assertions. What would you do if you were in the US/CS/UK General Staff's boots?
