As said above, Jacquetta (possibly) losing custody of her son if she remarries (one of the reasons that Catherine de Valois' desired match with the duke of Somerset foundered, and she chose Tudor instead "a match so low none can object thereto"), is actually unlikely IMHO. After the death of Queen Catherine, she's the highest ranked lady at court, she's aunt of the king by marriage and related to the new queen, plus, she's (until Marguerite d'Anjou comes along), England's only diplomatic link to the continent (with her ties to the house of Luxemburg, to the Valois and to the Holy Roman Empire).
I agree though, that with another coffin standing between York and the crown he might not get so uppity (though he still will some, I'm sure).
That said, the duke of Bedford's first wife died in childbed with a child of unknown sex (Italian wiki refers to it as "bambina" (fem), while another source says it was a son). Having that child survive PLUS one by Jacquetta could make things interesting.