First, to correct some names from Part 1. "Hooker" should read as Franklin, "the unnamed commander on the right" as Miles, and there are also Scheck's brigade and Willcox's brigade to throw into the fight.
Part 1 is assuming that, after the initial BP, fate is about even to both sides, or even helps the Confederates a little bit. Just as easily, it could happend that no significant Union general is wounded at all, Jackson is felled by close-range musketry, at least one of Beauregard and Johnson is captured, and so is Jefferson Davis (who we recall in OTL rode to the battlefield, insisting on seeing the battle), plus a few thousand Confederate troops prisoner and throw in another three brigades routed from the right. We are not asking for an extremely likely way for the Union to capture Richmond, simply one that is plausible.
But in any case, there will still be some 15-20,000 organized Confederate troops and someone in command of them, let us say Beauregard. At best he will have to fall back on Manassas Junction. There is no natural defense near Manassas Junction. The question becomes, where to retreat to? He has to retreat quickly, otherwise any aggressive move by McDowell and his 4 fresh brigades will easily rout what is left of the Army of Virginia/Army of the Shenandoah. He has essentially 2 options.
1 - retreat along the Orange and Alexandria RR, to the southwest. The main advantage to this route is that it is easy to maintain, and so prevents massive straggling/desertion. There are several disadvantages however:
It is out of the way of Richmond, and shoudl McDowell procede due south he would outflank Beauregard and go unopposed into Richmond.
It opens up the Confederate's left flank to attack from the Valley (more in a moment).
Most importantly, there is no obvious defensive line. The first reasonable location is at Warrenton Junction, behind Cedar Run. This is almost 20 miles from Bull Run, and there surely will be substantial desertions. Also, at the junction is the Warrenton Branch RR, where a line comes from the Shenandoah. By this point it will be clear that having Patterson in the valley is pointless as there are no Confederates there anymore. He could be recalled and joined to McDowell directly. More likely, he will advance down the valley to subjugate that area. He would able to get another 18,000 Union troops easily on Beauregard's left flank, virtue of the RR.
One final bad point, since it is along a RR, it will be easy for McDowell to receive supplies.
2 - retreat due south along backroads. It will be hard to supply either army on this route, and there are great opportunities for desertion.
However, Beauregard will probably take this route because of its advantages - it places him squarely between Washington and Richmond, and it has some semi-defensible river lines. The first, Cedar Run, is here only 7-8 miles south of Bull Run. It is another ~18 miles to Chopawamsic Creek, which has some possibility of being defended. Then it is 5 miles to Aquia Creek, similarly a delaying action. By this point the Potomac is on the Confederate's right, so they might retreat more slowly. However, retreat they must. They are outnumbered now easily 2 to 1 (more if Patterson gets moving) and demoralized. Plus, at this point Telegraph Road joins the retreating roads, so supplies can come direct to McDowell again, as well as heavy artillery. From Aquia it is about 10 miles to the Rappahanock, and Fredericksburg. Finally, a real defensive position.
The trouble is that even if Beauregard can successfully retreat so far without losing many supplies, cannons, and especially troops due to desertion, McDowell certainly will not. It will still be at best 2:1 ratio at Fredericksburg, and now McDowell has the rail-line to Aquia for supplies. He can wait there for a month if need be, until Patterson arrives, at which point his numerical superiority will reach 3:1.
Considering that Grant had 2:1 in 1864, that Lee actually did receive re-inforcements at strategic locations (North Anna), and that in 1861 there are virtually no prepared defensive works even at Richmond, if McDowell cannot capture Richmond within 2 weeks of reaching Fredericksburg, he must be a combination of all the worst aspects of McClellan, Pope, Hooker, Burnside, Bragg, Polk, Ewell, Hood, Buell, and a few dozen other bad generals.