It depended largely on the reigning monarch's reaction.
A tiny bit earlier than you wanted, Catherine de Valois (1401—1437) was the widow of Henry V of England (1386—1422), lionized in his own time as the victor of Agincourt, and mother to Henry VI (1421—1471).
She entered into a sexual, quite likely non-marital, relationship with one of her household staff, Owen ap Tudor (temp. Henry V & VI 1392?—1461), and bore him six children, among them Edmund Tudor (posthumous father of Henry VII) and Jasper Tudor (Henry VII's lifelong general and guardian).
Once Henry VI learnt of his siblings, presumably immediately before his mother's death (N.B.: he was a teenager at his mother's death, and entirely under the influence of his regent, and separated from her for most of his childhood), he freed his stepfather from imprisonment, and eventually raised his Tudor brothers to the peerage. Henry VI gave his cousin and Lancaster heiress Margaret Beaufort in marriage to his brother Edmund Tudor after he'd been raised to Earl of Richmond and reached adulthood. She was the mother of Henry VII.
However, Henry VI was widely condemned for being entirely too merciful to those who threatened his throne in one way or another (and it didn't help that he was insane, possibly with schizophrenia, either), and strengthening the claims of his bastard half-brothers wasn't viewed positively by all. However, this was done long before his first bout of insanity, and the Tudor brothers never threatened his rule, but instead supported him in the Wars of the Roses.
Henry VI's granddaughter, Margaret Tudor (1489—1541), sister to Henry VIII, is a more interesting case. She was wed to James IV of Scotland (1473—1513), who died at the hands of her sister-in-law, Catalina de Aragón, queen to Henry VIII, at the Battle of Flodden. During her marriage, she had borne a son who survived infancy and eventually became James V of Scotland (1512—1542). James V was one year old at his father's death, and his mother Margaret Tudor remarried a Scottish noble in 1514, Archibald Douglas, Earl of Angus.
In re-marrying, she was deprived of the custody of her son by her husband's political enemies, and her marriage did not last long, with her fighting consistently to get a divorce by 1518 (Angus was no saint, given that he'd taken a mistress that he was supporting on Margaret's money, and she'd spent a year living in England on her brother's support by that point), and eventually being granted a divorce by the Pope in 1527, and remarried again in 1528. Her brother, Henry VIII, had strongly opposed her petitions for divorce, his own desires for a divorce being long in the future.
Ironically, Angus kept custody of his stepson during the final two years of his marriage before fleeing into exile until his stepson's death. James V had suffered greatly for the marital choices of his mother, and after her marriage to Angus, she never regained her lost influence.
Further reading:
On Catherine of Valois and Owen ap Tudor:
Alison Weir, The Wars of the Roses, specifically the chapter on "The Child King". It's a wonderful book and a well-paced read, though biased towards the Yorkists.
Dan Jones, The Wars of the Roses: The Fall of the Plantagenets and the Rise of the Tudors.
On Margaret Tudor:
Maria Perry, The Sisters Of Henry VIII: The Tumultuous Lives Of Margaret Of Scotland And Mary Of France.
Alison Weir, The Lost Tudor Princess: The Life of Lady Margaret Douglas.