I think the first question is whether secession would have occurred as it did historically. Douglas was not the first choice of southerners, but would his election have made southerners feel like the crisis was upon them the way Lincoln's did? Especially since he had chosen a southerner as VP. Even those who thought separation might become necessary might be willing to wait and see if the administration actually did anything they considered threatening.
It would be an interesting situation if one or more states seceded but Georgia was unwilling to with one of its own as vice president.
But getting back on topic, assuming Georgia secedes and Douglas dies on schedule, the Unionist position was that secession never actually legally happened, regardless of proclamations made or acts of insurrection. There would therefore be no constitutional basis to say that a Georgian couldn't be president.
President Johnson could be impeached for treason or high crimes or misdemeanors if he was not considered to be prosecuting the war effectively. Conviction requires the concurrence of 2/3 of senators present, not 2/3 of all senators, so the absence of southern senators would not be an issue.
Prior to the 25th Amendment, there was no process for appointing a new Vice President; the office remained vacant until the next election. For example, the last President Johnson (Lyndon) had no VP until he and Humphrey were inaugurated January 20, 1965.
The Constitution provides that "the Congress may by law provide for the case of removal, death, resignation or inability both of the President and Vice President" but apparently this was not done prior to the 25th.