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.
How much sooner does the war end in this scenario?
It might even last longer. For all of McClellan's faults as a battlefield general, he was superb when it came to logistics and training, and he'd be unlikely to take command of the Union forces in the Virginia theater until at least 1862 if the Union wins Bull Run. Also, a neutral Tennessee would have all sorts of butterflies for how soon, or even if, Grant rises to prominence. But even assuming that other than Forrest beginning his military career in a Mississippi regiment, no significant changes in leadership happened, then it ends at most a year earlier in my opinion.
The Union winning Bull Run is unlikely to lead to permanent Union control of central Virginia, and even if it did, it won't make a significant change in relative military capability. An early loss of Richmond is what would make possible a significantly shorter war, not a change in Bull Run.
