Given the state of both armies that early in the war, even a decisive Union victory at Bull Run in 1861 would have left McDowell in no condition to pursue as the rebel Army of the Potomac skeddaddled off to Richmond. So in no way does it bring the end of the war in 1861. With so significant a change, it's doubtful Jackson gains his "Stonewall" sobriquet, and if he does, maybe General Bee can explain if he meant it as a compliment or insult.
So any Confederate collapse as a result of Bull Run would be political, not military, in nature. As of Bull Run, Tennessee had seceded but not yet formally joined the Confederacy. So it's possible that Bull Run leads Tennessee to attempt a Kentucky-like neutrality.
My impression that one of Bull Run's biggest impacts was psychological, convincing the South that they actually had a real shot at this. The fledgling CSA losing the first battle of the war might convince the CSA that the secessionist cause was doomed way earlier than that sank in in our timeline.
IOTL the effect of the Union defeat at Bull Run was to harden Northern resolve —why wouldn't the reverse have a similar effect in the Confederacy?
