I think people are underestimating Jane's power in this TL. She has done something that no other queen has done for Henry: She has given him a legitimate son. Not only one son, but she has given him two additional sons. She becomes tremendously powerful, and she can throw her support behind 1) Returning England to the Vatican and 2) Legitimizing Mary. Henry will definitely have small affairs, but Jane's position as Queen is unshakable. Before delivering Edward, she walked on eggshells around Henry in fear that she'd be killed or replaced, but no longer. And she's not the timid dormouse most historians portray her as: Even Chapuys, who was in her favor, called her proud, and Jane dismissed several ladies for not adhering to her strict how-many-pearls-on-a-bodice rule she made up, and she threw out women who wore Anne Boleyn's favored hoods. I like to imagine that Jane will live into her 70s as a powerful queen mother.
Now, I really don't think England can return to the Vatican (at least during Henry's rule), but Jane can influence her son to be Catholic. Jane can handpick Edward's tutors and teachers and ensure that they are good Catholics; when in the OTL, the closet Protestant Catherine Parr did. Edward was very intelligent, rigid, and passionate. In the OTL, his zeal was directed towards Protestantism. It was under Edward VI's reign that Protestant England became guaranteed: He forbade Mass, the Eucharist, and heavily persecuted Catholics. , but Jane could shape him so that he's as fervent a Catholic as Mary. In this TL, I think Edward would essentially maintain all Catholic traditions but retain his title as head of the Church of England and marry a Catholic princess.
The only reason Jane became queen was her conspiracy with Chapuys (who was Catherine of Aragon's BFF and #1 fan) and Catherine's old friends, who schooled her on how to seduce the king and make him want her. After Edward's birth, it's guaranteed that Jane will persuade Henry to legitimize Mary.
In 1538, Spain and France signed a pact of alliance and the Pope excommunicated Henry, so he is very anxious of being cornered by two great Catholic powers. He'd use his newly legitimized daughter to get a way in with one of these countries. At this time, Mary is only 21 and plenty young enough to have children. The newly legitimized princess is now Henry's only legitimate daughter and has ties to Spain and the Emperor. No way she's marrying locally, a mere English lord.
Henry would either throw in with France or Spain and offer his adult daughter to protect England's interests. Cromwell will beg Henry to marry Mary to a Protestant, such as the brother of the King of Denmark (who is Lutheran). Or, Duke William of Cleves, the brother of Anna of Cleves.
Chapuys wants Mary to make a marriage to help the Emperor-- maybe Dom Luis of Portugal, one of the King of Hungary's sons, or the Duke of Savoy's son. This is what Mary would prefer. She identified more with her Spanish mother than with her English mother, and she wanted to marry one of her Spanish cousins. Mary isn't Queen of England, so the Emperor wouldn't marry her to his own son Philip II.
OR. This is a very, very fun idea. Let's say Mary is legitimized in 1538, a year after Edward's birth. In 1541 in the OTL, King Francis I of France and Norfolk were negotiating for one of Henry's daughters to marry his second son, Henry, Duke of Orleans; the Duke was married, but Francis was confident that he could dissolve the marriage since Catherine de Medici seemed barren at the time (and she wouldn't have a child for the first 10+ years of her marriage). The negotiations fizzled out since Henry refused to legitimize either of his daughters. In this TL, the 22-year-old Princess Mary Tudor could marry the French king's son. When the Dauphin dies before siring a son, Princess Mary will become the Queen of France. Since she is still very young and she is much happier, she could have children. In the OTL, she started trying to have children at 38 and she had suffered deep depression under Edward VI's reign as he brutally suppressed her Catholic faith. Mary never wanted to be a queen regnant, and I can see her very happy in France, even if she'd at first be cold to her husband, since the Valois are the enemy of her maternal relatives, the Hapsburgs. She'd support her husband's persecution of the Protestants.
Jane won't lift a finger to help Elizabeth. She ignored Elizabeth and fawned over Mary. Without a Catherine Parr to champion her, Elizabeth probably won't amount to much. Henry'd probably try to use Elizabeth in the international marriage game, but she is A) a bastard and B) behind her 3 legitimate brothers and sister in the succession, so few kings or princes would want her. In the OTL, Henry tried to marry her off to a Denmark prince, but the King of Denmark scoffed that he wouldn't marry his brother to a bastard, leaving Henry baffled. She'd probably marry a local English nobleman. Robert Dudley won't be the Earl of Warwick, since his 2 older brothers will live. Despite her bastardy, Elizabeth is a king's daughter and will be well-dowered and a prize for any local Englishman. It's possible that Jane will convince Henry to give Elizabeth as a "prize" for one of her Seymour relatives, so Elizabeth might end up marrying Thomas Seymour, since Edward is associated with scandal after his incestuous, cuckolding wife.
Jane Grey lacks importance. Henry has 3 healthy sons and 2 daughters, one of whom is a bastard but still is before Jane in the succession. She will lead a happy, if unremarkable life as a Protestant scholar. Her ambitious parents the Grey will no doubt marry her off, but she won't be the 9 day queen.
Who will Edward marry? Well, perhaps Mary, Queen of Scots. But if Mary Tudor DID marry King Henry II of France, the French wouldn't be so eager to snatch the infant queen away for their Dauphin (assuming Mary has a son by this time). Or, if Mary doesn't marry Henry II, then Edward could marry Elizabeth de Valois as he was betrothed to in the OTL. He could always marry one of Emperor Ferdinand's daughters or granddaughters.
Honestly, the marriages of Henry and Jane's sons depend too much on whether Edward will stick to Protestantism or Catholicism. If Protestant, German princesses from the Protestant League are most promising (still, at least Edward or his next brother will try to marry the Queen of Scots), since there'd be a religious conflict in Protestant/Catholic marriages.