In essence, Lee has the ability to completely destroy the entirety of Meade's command.
What's going to happen next?
On the political level, it's important to note that the 55,000 casualties suffered by Grant were enough to so shake Northern morale that Lincoln until the end of August thought he was going to lose re-election; Lee here in the ATL has done that better by increasing the losses by a third and completely destroying the chief Federal army. While Lincoln is blessed that this great defeat is not with his re-election at hand, it still comes during Congressional and State-level races that even IOTL saw, for example,
a Copperhead endorsed by McClellan come within a hair of winning the Governor's office in Pennsylvania. This defeat also comes when memories of the New York City Draft Riot,
Detroit Race Riot and
outright battles with Draft resistors in Ohio are still fresh; we're also mere months away
from disturbances in Lincoln's home state of Illinois. The Peace Democrats are going to be incredibly strengthened by this, with major repercussions going into 1864.
At the strategic level, however, things are even worse. Outside of the troops in the immediate environs of Washington, there is no real Federal force to oppose Lee in the Mid-Atlantic. There are also no real prospects for any such force being constructed soon, as the Lincoln Administration had stripped what surplus forces existed in the aftermath of Chickamauga in order to rescue the now besieged Army of the Cumberland at Chattanooga; Grant had detached 20,000 under Sherman, while Hooker had been sent with 15,000 from the Army of the Potomac. Pulling out any other forces thus opens up serious dangers in other theaters, which greatly constrains the options for the Union cause. In short, Lincoln will have to decide whether he wants to save D.C. or, most likely, see another great military disaster around Chattanooga.
If the Federals fail to move to protect Washington, Lee will move to occupy Centreville and emplace batteries along the Potomac,
closing it down to riverine traffic just as the Confederates did for nearly a year back in 1861-1862. Lee can then take the majority of his host, swing into Maryland and then occupy Baltimore, closing off the only rail connections into Washington. With the railways and the Potomac closed, the city will inevitably be forced to surrender at some point. In the meantime, with Lee in Baltimore, a secession convention can be organized for Maryland. Between the decisive defeat of Meade, D.C. under Siege and Maryland now in the Confederacy, it's such a disaster for Federal arms that French intervention becomes essentially assured.
Should the Lincoln Administration attempt to save the city, the most likely route for such would be to pull the 15,000 troops of Burnside's Department of the Ohio out of East Tennessee. Likewise, the 15,000 men that had been detached for Chattanooga from the Army of the Potomac under Hooker could be withdrawn, likely arriving in less than two weeks. 30,000 men isn't enough to take on Lee in the field, but it's sufficient to strongly picket the entry points into Maryland and reinforce Washington. Combined with Lee's hesitancy to campaign that far North with winter coming, it's probably enough to deter the Confederates. The problem is, however, that it opens up the Federals for certain disaster elsewhere, Chattanooga in particular.
Without Burnside at Knoxville, there's no need for the Knoxville Campaign, opening up the railway network for use in aiding Bragg's logistics. Perhaps equally important is that the 10,000 men under Maj. Gen. Samuel Jones in Southwestern Virginia can now perform a link up with Bragg's Army. Returns from November suggest this would mean 69,000 Confederates against 57,000 Federals, once you remove Hooker's 15,000 from the total. However, the Federals have a further disadvantage in that around 40,000 of their number is trapped within Chattanooga, on the brink of starvation in late October. These facts alone make clear that Grant is in serious danger here before you even consider specific operations.
Case in point is that Hooker's men were used to protect Bridgeport, Alabama and its connections to Chattanooga, meaning what would become the "Cracker Line" origins point is dangerously exposed. Further, without Hooker's men to guard Wauhatchie, Longstreet can take and directly cut the Cracker Line as its main point in Chattanooga. No matter which way you look at it, it's definitely likely the attempt to relieve the city is going to fail. Accord to Thomas, in his famous correspondence with Grant when the latter arrived on scene, the Army of the Cumberland had, at most, seven days worth of rations left. By the time the Cracker Line was opened IOTL, they had, at most, a day's worth. No matter how resolute George Thomas is, when the supplies run out it's only going to be a matter of a few days before the Army of Tennessee is able to bag the 40,000 men of the Army of the Cumberland. Thereafter, with only Sherman's 20,000 on scene and no hope of reinforcements, Bragg can either destroy the remainder of the Federals or, much more likely, retakes Tennessee and sets himself up in a position to move into Kentucky come Spring.
Between all of this, I think it's safe to say Northern willpower to carry the war is going to be dangerously depleted if not outright destroyed. Even ignoring that, the French under Napoleon III were still serious about intervention into the Fall of 1863 and these decisive Confederate victories make such a move a near certainty. On the whole, I think this is a war winning scenario for the Confederates.
Bringing the