For the sake of AH, i'm going to go with Grey Wolf's scenario.
Abraham Lincoln would not be leading the (let's call it) 4th Continental Congress. It would most likely be led by William H. Seward and Stephen A. Douglas.
Lincoln would most likely attend the convention, but he'd probably be more of a Madison figure: a rising star, but not experienced enough yet to be given control.
Other attenders of the convention would most likely be James Buchanan (who would be championing compromise with Britain against war), John Breckenridge (who would be representing his constituents in the south), Edwin Stanton (championing war), Andrew Johnson (assuming he doesn't stick with tailoring ITTL), Benjamin Judah (who might bring the concept of how minorities would play in the upcoming election), Jefferson Davis, Salmon Chase, Howell Cobb, Thaddeus Stevens, Charles Sumner, and probably a few other smaller figures, but those are probably the major ones.
Of course, assuming their lives go relatively similar in British North America as they did in the United States.
This Continental Convention would probably be seeking a military leader as well: Robert E. Lee might be at the top of that list in the same way that Washington was in 1776. George McClellan, Albert Sidney Johson and Thomas Jackson could be on that list as well, but i think Lee would end up commanding it by the end of the war. Him and Grant (i think given virtually any military circumstances that doesn't give the British overwhelming technology or military strength) would most likely end up as the generals that the war ends up with as their big heroes.
The big issue i think would have to be slavery. It makes or breaks BNA's struggle for freedom. Did William Wilburforce succesfully establish manumission in 1833 and the southern colonies of BNA accepted it? Then this problem would be nonexistent. Or did the colonies manage to escape Wilberforce's reign? At which point, this is a serious issue.