Reposting his full question:
If the Confederacy had become independent, would their english eventually be considered a different language? (Very similar of course, like the relationship between Dutch and Afrikaans). One could assume the prestige dialect would be as distinct as possible from “Yankee” speech, there would be much less media/cultural influence over southern english to ensure that it was relatively intelligible to Northerners, the little to no immigration to the Confederacy would ensure the North would drift further away, and most importantly, the Confederacy would have had an army and a navy.
If this was 1000 years ago, sure, they would have drifted apart. But 150 years apart in modern times? With the universality of print (including print from the UK)? I think you’d get a situation more like Australian English vs British English. The prestige accent certainly wouldn’t be Midwestern in the CSA, accents would diverge a bit more, and you might see idioms like fixin’ to in standard CSA English which you won’t in standard US English.
But I believe the forces that have kept US English and UK English mutually intelligible would still in play for US and CSA English, even if they hated each other.
EDIT to respond to Jason Blau’s question in comments.
Spoken English dialects? With less Damn Yankees around, with a less industrialised economy so less mobility in general, and with less of a centralised identity pushed in schooling (it is the con-federacy after all)… there’d be more drift, yeah. Not sure if that would extend to the Bayou though: it’s still a “foreign” language, and I can’t imagine that there’d be no Speak American sentiment in the CSA (or rather, Speak Southron).