The overwhelming of Hastings host happens as it does in OTL, but instead of the men under Oxford looting Hastings' men, they remain in order and rejoin the battle spreading word of Hastings death and the fall of the Yorkist left wing. This crushes morale, in the Yorkist host and the added presence of Oxford and his men helps turn the tide in the clash between Montague and Edward, with Oxford and his men shouting to the heavens about the death of Hastings, the Yorkist forces begin to grow tired and broken, smashed between the hammer and the anvil, the Yorkist men. Broken and somewhat disheartened, the men under Gloucester's command capitulate under forced pressure of Warwick and Oxford's hosts. Breaking the Yorkist force and winning the battle of Barnet.
With Gloucester slain and Edward IV captured, the Yorkist campaign for the throne is all but at an end. Margaret of Anjou and her son Edward of Westminster march with their army to take London, Edward IV's sons Edward and Richard are imprisoned in the Tower of London and Edward's father Henry VI is freed from his cell in the tower and reinstated as King.
Henry VI would only survive for about a month more and then would die during his sleep leading to Edward being crowned as King Edward V. The issue of what to do with Edward IV's children as well as fixing the realm after the war continues to dog Edward through most of his early reign, a problem that is only worsened when Princes Edward and Richard go missing from the tower in 1478.
First, Barnet was in April 1471, Edward IV only had
one son at that point (OTL Edward V) born in the November of the previous year. Second, Margaret and Edward of Westminster only
landed in England that Easter weekend and basically the first news they heard was of the defeat at Barnet.
But, let's assume the events progress as you've suggested: Edward's as good as dead the minute he's captured, and he would know it. He's rebelled against his lawful king, and in this case, lost. And he'll no doubt be beheaded as a traitor and made a fearsome example on.
However, even if he is, Queen Elizabeth (probably Dowager Duchess now) has given birth to a son in sanctuary at Westminster. The Lancastrians (the church party) wouldn't be so keen to violate sanctuary as Edward himself was, but I personally don't see the kid living very long. Somehow, the Lancastrians will be determined to get rid of him.
Of course, Clarence isn't exactly home free either. Technically, with Edward IV dead and the young Edward V dirtying diapers, he's
de facto head of the Yorkist party. And while he's pro-Lancastrian at this point, that
can change. How, I'm not sure, but I should imagine the Lancastrians can drive him to it, simply by Edward of Westminster having a son or two, thereby displacing Clarence who was named as the heir-presumptive to the Lancastrian claim with the readeption. Somehow I'm thinking all that's changing in Clarence's future is that it might be a different person's signature on his death warrant.
I think Gloucester's words from
The White Queen are significant "Hush, George, don't forget, you went down on your knees for Warwick when you thought he could make you king. You're as much a traitor as he.".
That said, Warwick and Margaret of Anjou is an alliance born to fall apart. It's all well and good while she doesn't really have any other choice, but Warwick's going to be far from the only (or most astute) political player at the Lancastrian court. He's going to have the stench of traitor about him, and it's going to affect anything he says/does, even if he is EoW's father-in-law.
Which brings us to the Princess of Wales - Anne Neville. Her obstetrical history with Gloucester was shaky at best (one kid), even if we assume that there were more failed pregnancies that we don't know about. Her sister produced four children of whom two survived (and if young Warwick hadn't been executed, there's reason to believe he might've lived as long as his sister, the Countess of Salisbury), her mother only two. Granted, she's further related to EoW than Gloucester (third cousins rather than first), so she might have at least two. That's of course assuming that MofA keeps her around long enough (since it's questionable if the marriage was solemnized). And a Prince of Wales who
will succeed is in a very different marriage situation to a prince of Wales-in-exile who
might. AFAIK, there were considerations for the Princess Margaret of Scots at a point, possibly an annulment might be forthcoming from Rome (for whatever reason), and those plans dusted off.
Lastly, EoW is not going to be crowned Edward V. That would be acknowledging EIV was a legitimate king, and that's going to be a difficult pill for even the most die-hard Lancastrian York-apologist to swallow. I suspect he might be vilified much like Gloucester was by the Tudors. So, he'll be crowned as Edward IV not V.