Why would Charles of Burgundy go Lancastrian? He has zero to gain by it. He's married to the sister of the Yorkist King, and any chance his issue has at pressing a claim to the English throne is by virtue of the Yorkist connection.
I also think Jacquetta of Luxembourg would laugh Margaret of Anjou out the room if she tried to turn her - now exalted and vindicated as the matriarch of a powerful faction, mother of the Queen and grandmother of the future King - against the Yorkists on behalf of a Lancastrian newborn in Yorkist hands and betrothed to the said future King her grandson. Jacquetta wasn't married to Bedford all that long and had nothing to gain risking all on behalf of her late husband's great-great-niece (who could well die in infancy or marry abroad once placed on the throne, scuppering any Woodville-involving plans).