Union Victory at Fredricksburg?

What if, at the Battle of Fredericksburg, in December of 1862, the Union captured it's objectives without any issue?
The Plan under General Burnsides was to use Pontoon bridges to cross the Rappahannock River and captured the town of Fredrick Virginia and the Rail line to Richmond, the CSA's capital, which would have forced Lee into an open battle.
But due to administrative bungling in Washington, they arrived fives days late, meaning Lee had already had time to move his Army to the Rappahannock's shores and set up a battle plan, this is, in my mind, what lead to the Union's defeat.
But what if it hadn't? What if, Lincoln, getting wind of such a error ordered them sent forwards? Or perhaps the Army put aside it's misgiving and mistakes and the plan did work, the Union crosses the River and takes the crossing with Richmond less then 100 miles south by rail? What happens now?
 
If the pontoon bridges arrived at Fredericksburg on November 17/18, then Sumner’s Grand Division (and Franklin’s on the 18th) would be able to capture Fredericksburg before any meaningful Confederate opposition be made to resist the crossings. The rest of Burnside’s army would arrive on the 19th of November.

IOTL, the first Confederate force to march to Fredericksburg was two divisions of Longstreet’s Corps and W.H.L Lee’s cavalry brigade on November 18. The first Confederate force to reach Fredericksburg was McLaw’s division, which arrived on the 20th of November, followed by Ransom’s division (22nd November) and the other 3 divisions of Longstreet’s Corps on the 23rd of November. Jackson’s Corps left for Fredericksburg on November 21.

Lee would still intend to confront Burnside’s advance south, Lee had not expected to reach Fredericksburg first and had intended to fight on the North Anna. Given that Lee would know when Burnside had crossed Fredericksburg thanks to his scouts, Lee would have likely redirected Longstreet’s and Jackson’s Corps towards the North Anna junction.

Although Burnside has reached Fredericksburg first, I’m not certain if Burnside would immediately march towards Hannover Junction. While he is aware of Lee’s old postions, he might not be aware of Jackson’s position after he crossed the river. He may be cautious and wait until he collects enough intelligence to determine Lee’s positions.

Regardless of where the battle with Lee is fought, Burnside is likely forced to enter winter camps after the battle. The weather was becoming rapidly unsuitable for continuing offensive operations, unless Burnside attempts a Mud March and get bogged down.
 
The bigger question has to be, why do they need the pontoons to cross in the firsthand? If the AotP becomes aware of just how critical it is for them to get onto Maryes Heights and through the town, they will cross without the pontoons and merely use a grand battery of artillery to cover the army on the other side. One Confederate division is not going to take on a dug in Union Grand Division, even if it is without its baggage.

Had Burnside gotten into Fredericksburg, however, he would not have been able to push a rush on Richmond. Lee would be denied using the Rappahannock as his line, but that is still a maneuver victory and does not necessarily need to be pushed further. The armies likely would be looking at each other from across the North Anna over the Winter (the Mattapony River Line would be easily flanked, and the ground in the area was too open for Lee to want a slugfest of a battle). Burnside would then be faced with a choice. He could try to cross the river head on (there were many fords, and Lee's defenses could probably be flanked somewhere), or, he could rather try to march on Gordonsville in an effort to spread Lee out before doing anything decisive. I think its more likely he tries the first option than the second, which would take a lot of coordination and complication that he might not want any part of.

From there, its hard to say what happens.
 
The bigger question has to be, why do they need the pontoons to cross in the firsthand? If the AotP becomes aware of just how critical it is for them to get onto Maryes Heights and through the town, they will cross without the pontoons and merely use a grand battery of artillery to cover the army on the other side. One Confederate division is not going to take on a dug in Union Grand Division, even if it is without its baggage.
I must disagree. While I would agree that Burnside could have ordered Sumner to capture the town of Fredericksburg itself, without a pontoon bridge Sumner cannot cross in large enough numbers to occupy Marye’s Heights, just Fredericksburg itself.
Burnside would then be faced with a choice. He could try to cross the river head on (there were many fords, and Lee's defenses could probably be flanked somewhere), or, he could rather try to march on Gordonsville in an effort to spread Lee out before doing anything decisive. I think its more likely he tries the first option than the second, which would take a lot of coordination and complication that he might not want any part of.
I would agree with you that Burnside would likely try to push across the river, a march to Gordonsville from Fredericksburg is completely impossible to execute with logistics in consideration. However, I would have to disagree with the idea that Burnside would have flanked Lee’s defenses on the North Anna. As Lee showed in 1864, North Anna was quite a defensible position.
 
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