The Battle of Barnet was fought on the London Road roughly a half of mile outside the town of Barnet. Edward Hall reports that Richard Neville the Kingmaker, 16th Earl of Warwick, had hoped to beat King Edward IV to London but reports of the Capital’s fall had reached the Army on April 12th 
. Knowing that Edward IV was marching from London and that he would seek a decisive confrontation, the Kingmaker sought a secure and defensive position to withstand the Yorkist attack. The battlefield was carefully selected as the Lancasterian Army was arrayed along a ridge with a swampy forest further securing their forces from a potential encirclement 
. At the Council of War, it was decided that the Earl of Oxford would command the left flank while the right flank would be commanded by the Duke of Exeter. This meant that the Neville Brothers would command the Center against the expected attack from Edward IV. Sensing the low morale of the Lancastrian Army following the news of the fall of London, Marquess Montagu convinced his brother to fight on foot among the soldiers.
Despite having chosen the battlefield carefully, Warwick was not prepared for the speed of response from Edward IV, who left London at dawn and arrived at Barnet as night was falling on April 13th. The Yorkist camp was significantly closer to the Lancastrian line that was expected. As a result, the Lancastrian artillery barrage was ineffective while the Yorkists didn’t return fire to avoid revealing their position.
On April 14th, the Yorkists began their attack aided by a “blanket of fog,” which restricted the field of vision to a few yards and prevented a proper Lancastrian artillery barrage. This weather would play a key role in the engagement to come as the various divisions of the Armies quickly became disengaged. The most significant developments of this battle happened along the flanks as the Yorkist right flank led by the Duke of Gloucester were able to outmaneuver the Duke of Exeter and begin a genuine assault on their flank. Had the Duke of Glouchester not been fighting up an incline this would have had the potential to roll the Lancastrian line but this minor delay enabled the Lancastrians to rally and prevent the Left Wing from collapsing.
While the Yorkist right was making progress against the Lancasterian left -- The same can not be said for the Yorkist left under Lord Hastings. Lord Hastings had been among the most loyal and skilled of Edward IV’s lieutenants but as a result of the fog, he had misjudged the Lancaster Right leaving his flank exposed to a counter-attack. A counter-attack that the Earl of Oxford was more than ready to deliver. The Lancastrian right attack was swift and without the disadvantage of the decline barreled into the Yorkist Left. The Yorkist left was quickly put to the flight after minimal resistance despite the efforts of Lord Hasting to rally his troops 
. Whilst trying to rally his troops Lord Hasting was stuck down amidst the chaos. Despite the Earl of Oxford’s attempts to keep his troops under control, the Lancaster right pursued the routing Yorkist forces into Barnet, where the proceeded to loot the city 
. It would require a significant amount of time before Oxford would be able to return to the battlefield.
At the start of the battle, both armies had expected the battlefield to be decided by who was victorious in the center but as a result of the pressure from Gloucester, Warwick and Montagu had been forced to swing their Army eastward whilst Edward IV had swung his army westward. The battle was nearing 3 hours at this point with minimal progress being made by either army in the center when the Earl of Oxford finally returned to the battlefield from Barnet. Expecting to attack the Yorkist Army from behind, he instead encountered Montagu’s division. Marquess Montagu nearly ordered his archers to fire upon Oxford’s division having mistaken his heraldry of a ‘Star with Rays’ for Edward IV’s ‘Sun in Splendor’ but fortunately before the order was issued, the Earl of Oxford was spotted among the men 
. The reinforcement of the Earl of Oxford would greatly reinforce the Lancastrian Center as Edward IV had made a daring attack and committed his reserves and personally attacked the Lancastrian Center. The fighting was fierce but it seemed like the Lancastrian Center would repulse the attack when news reached that the Duke of Gloucester had successfully put the Duke of Exeter to flight 
Warwick fearful of a rout began to order that the Lancaster Army begin an organized retreat using the cover of fog that was only beginning to lift to their advantage. The rearguard action was entrusted to the Marquess of Montagu, the most skilled Lancastrian general. According to a Burgundian observer that Marquess of Montagu was still fighting on foot like ‘a courageous knight’ and was ‘cutting off arms and heads like a hero of Romance’. Yet for all his skill in battle, he would eventually be slain. Despite the death of Montagu, he was able to keep the bulk of the Yorkist Army occupied long enough for a significant portion of the Lancastrian Army to successfully escape North 
. By the end of the day, it was clear that the Yorkist Army had won the day but a significant portion of the Lancastrian Army had escaped in good order.
Death of John Neville, 1st Marquess of Montagu
Analysis of the Battle of Barnet
Both Edward of York and Warwick had sought a decisive confrontation that would put an end to the War of the Roses but the Battle of Barnet ended up being largely inconclusive. It would be accurate to describe the battle as a Tactical Yorkist Victory as Edward of York controlled the battlefield at the end of the day but a Strategic Lancaster Victory as a vast majority of the Lancaster Army had successfully retreated from the field in good order.
The Battle is best remembered today for the death of William Hastings, 1st Baron of Hastings, and more importantly John Neville, 1st Marquess of Montagu, who was slain whilst commanding the Lancastrian rearguard.
London fell into Yorkist hands largely due to infighting between Warwick and Somerset
Theoretically, this position maximized the Lancastrian artillery
First POD - In OTL Lord Hastings survived the rout of his forces. He was among the most skilled and loyal of Edward IV’s lieutenants.
Same as OTL
Second POD - In OTL, Montagu ordered his archer’s to fire upon the Earl of Oxford’s men believing them to be Edward IV’s men. The Earl of Oxford subsequently quit the field but the allegations of ‘Treason’ broke the morale of the Lancastrian Army causing the army to break
Third POD - The battle has lasted longer than OTL allowing the Duke of Glouchester to defeat the Lancastrian left flank
In OTL - The death of Warwick and Montagu resulted in the complete collapse of this Army