Catherine Howard was probably the best positioned biologically to have a child.
How so? Just because she was the youngest? Or were you referring to her familial history of successful pregnancies into the mid-40s?
But as to the actual topic, this probably means that there is a massive cover up as to the whole Thomas Culpeper situation by the people who need her to be safe (i.e. the Howards). Let's say she becomes pregnant during the escape from the plague in 1540, giving birth in May/June of 1541 to a son, who we shall call Edmund to prevent confusion with his father and because I'm sick of Henrys and Edwards. Thus, little Edmund, Duke of York is probably given his own household out in the country and Catherine is safe. The Howards do more to protect her than OTL and Henry most likely demands the evidence against her, if it does come up, be disproved. Catherine, meanwhile, most likely lives a charmed life and might even give Henry a second child, although I'd say that's unlikely. However, it's when he dies that it gets interesting.
For one thing, Edward VI is most likely a much less enthusiastic Protestant. Catherine showed no signs whatsoever of being interested in Religion and a son isn't going to change that. Most likely she let's Henry keep her stepchildren on the same path while she dresses up and hosts all the royal events. Thus, even if we end up with the same situation again as OTL, Edward is not going to persecute his sister to the degree he did OTL and Mary most likely stays away from court to live a happier life as the spinster Princess and dies in 1558 as a Lady/Princess, maybe even living longer without the stress of ruling on her shoulders. Let's say she survives into around 1561/2. A few more years of nothing really.
Elizabeth, meanwhile, doesn't have the second major blow of Catherine Howard's death to destroy her psychologically and thus is more willing to marry. Even if she's still unwed at her first brother's death in 1553, she most likely married to some minor Prince overseas or to a Howard boy. Let's say no one wants the semi-bastard sister of the English King and she is married in the late 1550s to Thomas Howard (future Duke of Norfolk) during the regency of Edmund I of England.
Now Catherine isn't going to be a respectable widow, prancing around with no lover. Most likely she remarries, but to who. She might pull a Catherine de Valois/Jacquetta of Luxembourg and marry some handsome knight and have a bunch of children, but she might also go for some rich lord who'll continue to take care of her as she wants to be. I'd say, if Thomas Seymour is free (a possibility. Even if he married Catherine Parr, she dies in childbirth OTL so who's to say it won't happen again) I could see the two of them marrying and him being raised to maybe an Earl, if just to keep the prestige of the royal family high. But that's a little fanciful, so let's say she marries Sir NoName McNormal, with whom she has a few kids with and embarrasses her son somewhat at Court events. She's going to have lots of children, look at how her mother did in over 20 years of pregnancies.
Edmund is most likely not a highly religious boy, if he takes after his mother or sister Elizabeth. Let's say he's similar to his father, Church of England but not much else. He'd most likely be married to Elisabeth de Valois as he brother had been planned to be, so let's say they have 3 living children: the Princesses Catherine (b.1560), Mary (b.1563) and the Prince of Wales, Edward (b.1567). A miscarriage or two in between and a dead Queen by the time 1570 rolls around mean we have a young, virile king with a sure succession, a need for a new wife and a family history of marrying into English families. There is going to be a lot of push to get in his bed.
Now, how long does he live? Well, we could go with how every other son of Henry VIII of England did and say up until maybe 19. But that makes it boring. So let's say he inherits his health from his mother. Catherine herself was beheaded, but the Howards all seemed to live at least into their 50s. And his sisters both made it at least into their 40s, so I think 50 is the benchmark here. Elizabeth died almost 70, but Mary died in her 40s due to cancer. Let's say King Edmund makes it to his 60s, maybe 63? That sounds about right.
Back again with his second marriage, there is going to be a push to get in his bed. I'm going to rule out Howards just because we don't want a Hapsburg situation. Now there are the Seymours, but they seem to have been a lot of mediocre looking people, and non set the world on fire. Catherine Grey would have been a good bet, but she's either married or dead (or both) by this point. Lettice Knollys will be free by 1576, but then again she's a fairly close relation. So I say he goes around and sleeps with the Courtiers women by eventually marries abroad again, for prestige.
That means we have to find him a foreign bride. If we continue the French alliance and Mary, Queen of Scots isn't available, I can see him trying for Marguerite de Valois. But that's not going to happen, she's will either be marrying Henry of Navarre or Sebastian of Portugal. Renata of Lorraine isn't available, but her sister, the lovable Dorothea of Lorraine is. However, she never had children, most likely due to her deformities. Catherine de Bourbon is a little young, but is also a full Princess so she might do.
Now, on the Hapsburg side, we have to figure some stuff out. Without a marriage to Mary Tudor or Elisabeth de Valois, I can see the proposed marriage to Margaret of France going through. Thus, we might have a perfectly aged Spanish Princess up for becoming Edmund's second bride. Let's say, however, the couple have two sons and a daughter a little young for Edmund surviving. Let's call her, Isabella of Spain (b.1559). Now, without Margaret to marry Emmanuel Philibert of Savoy, I can see the Duke marrying some other French Princess, but who? Well, we might actually get him married to Catherine de Bourbon, as she is really one of the only French (kinda) Princesses available. However, let's say he marries Catherine of Guise and has a few kids.
So, ultimately, it is up to Edmund to chose between an alliance with Spain or France. I ultimately think he'd keep with the French, thus marrying the Protestant Catherine de Bourbon in 1574. His teenaged bride might never have children, but let's say she does and a second son and daughter are born, along with say 3 other pregnancies.
Edmund I of England (b.1541: d.1604) m. Elisabeth de Valois (b.1545: d.1570) (a), Catherine de Bourbon (b.1559: d.1604) (b)
1a) Catherine Tudor (b.1560)
2a) Miscarriage (c.1561)
3a) Mary Tudor (b.1563)
4a) Margaret Tudor (b.1565: d.1565)
5a) Miscarriage (c.1566)
6a) Edward VII of England (b.1567)
7b) Arthur Tudor, Duke of York (b.1576)
8b) Miscarriage (c.1578)
9b) Miscarriage (c.1581)
10b) Elizabeth Tudor (b.1585)
11b) Miscarriage (c.1590)
Catherine Howard (b.1524: d.1581) m. Henry VIII of England (b.1491: d.1547) (a), Sir William Tresham (c.1530: d.1570) (b)
1a) Edmund I of England (b.1541: d.1604) m. Elisabeth de Valois (b.1545: d.1570) (a), Catherine de Bourbon (b.1559: d.1604) (b)
1a) Catherine Tudor (b.1560)
2a) Miscarriage (c.1561)
3a) Mary Tudor (b.1563)
4a) Margaret Tudor (b.1565: d.1565)
5a) Miscarriage (c.1566)
6a) Edward VII of England (b.1567)
7b) Arthur Tudor, Duke of York (b.1576)
8b) Miscarriage (c.1578)
9b) Miscarriage (c.1581)
10b) Elizabeth Tudor (b.1585)
11b) Miscarriage (c.1590)
2b) John Tresham (b.1550)
3b) Mary Tresham (b.1551)
4b) Thomas Tresham (b.1554)
5b) Jane Tresham (b.1557)
6b) Susanne Tresham (b.1558)
7b) Hannah Tresham (b.1560)
8b) George Tresham (b.1562)
9b) Charles Tresham (b.1565)