Assuming a CSA-wins scenario without foreign intervention (no Trent incident war or TL-191 style UK-French mediation to end the war) both US and CS remain Atlantic powers. For the rest of the 19th century, that means both countries will remain tied to British (and to a lesser extent, French and German) capital; both will need to maintain good trade relations in order to build railroads and industrialise. Therefore, once German and British interests collide, as they almost inevitably must in the early 20th century, both North American nations would probably come in on the side of the Allies. Assuming the war still starts in 1914, the CSA would probably declare war in 1915 or so (a la Italy and the Ottomans), have its army frittered away by the British in Gallipoli-style sideshows, with the USA following more reluctantly in 1917 or so for the deathblow against Germany on the Western Front. Cooperation in the war (especially in contrast with the European powers) might lead to a more permanent alignment of military, economic and diplomatic interests in the Old World between the two powers.