So in this scenario, Alt-Henry Tudor (probably called Margaret) is either married to a Grey or Richard III of England. It's more interesting if she becomes Queen, so let's go with that.
Scenario: The Lady Margaret Tudor, only daughter of Margaret Beaufort, was one of the greatest heiresses of her time when she married the Duke of Gloucester. Her mother, had been married to Sir Henry Stafford, a son of the Duke of Buckingham, with the understanding that their potential son would inherit her fortune and possibly, if their fortune held, the title Duke of Somerset. However, as of 1470 no son had been born, and Lady Stafford's daughter, Margaret Tudor, was betrothed to the King's youngest brother, Richard, Duke of Gloucester. They were married in January of the next year, when the bride was not quite 14 years old. The original marriage contract had specified the bride be older and more physically mature, as the Lady Margaret shared her mother's physical smallness, but the death of her stepfather had led to her mother demanding the marriage take place earlier, to protect her child.
The Duke and Duchess of Gloucester led a blissful life for the first few years of their married life. The Lady Margaret, physically tiny, darker coloured and witty, stood out amongst the tall, fair Grey women of the court, all of whom shared a similar type of pleasantness. With her strictly religious upbringing, and her quickly rising intelligence, Margaret found a place in the royal family, easily befriending the King and Queen.
The first child for the Duke and Duchess of Gloucester, a son, was born in 1473, when the bride was 16 years old. The child, named Edward after the King, was the first of five children born over the few decade, with three sisters and a son following in 1477 (Cecily), 1478 (Richard), 1481 (Anne) and a pregnancy in 1483 (Margaret), when Edward IV of England died, leaving his son as King.
It is not quite clear what led to the Duke and Duchess of Gloucester to make the moves they did to become King and Queen. The apparent death of Edward V of England in the Tower of London, alongside his brother, seems to have been an impetus, but later historians would make the argument that Richard III of England might have actually had the boys killed, although some have lain the blame of the team of his ambitious wife and mother-in-law. Granted, the two Margarets were pillars of the foundation that held Richard on the Throne, but ultimately it was his own actions that brought him there.
It seems to have Margaret who demanded that the Edward of Clarence and his sister Margaret remain in custody throughout their adolescence following rebellions in Edward's name. An odd move from the previously politically inactive young woman, it may have been that Margaret saw their potential for bad mental health, as both, particularly Edward, were known as for their odd antics later, when they were allowed visitors.
The marriage of Edward of Gloucester, Prince of Wales to Elizabeth of York also seems to have come from the Margarets. Having had the former Princess stay with Margaret Beaufort, the marriage of 7 years age difference seems to have not been happy for either party. The Prince of Wales was an unhealthy boy who died young, in 1494, leaving a son and daughter. However, the marriage did it's job in combining the bloodlines and ending any thoughts that another line could inherit, particularly after all of the York Princesses but Cecily of York were sent to the church, where they languished until their various deaths.
Richard III of England (b.1452: d.1503) m. Margaret Tudor (b.1457: d.1515) (a)
1a) Edward, Prince of Wales (b.1473: d.1494) m. Elizabeth of York (b.1466: d.1532) (a)
1a) Richard IV of England and Ireland (b.1491)
2a) Mary Plantagenet, Princess of England (b.1493)
2a) Cecily Plantagenet, Princess of England (b.1477: d.1507) m. James IV of Scotland (b.1473: d.1513) (a)
1a) Margaret Stewart, Princess of Scotland (b.1493)
2a) James Stewart, Duke of Rothesay (b.1495: d.1499)
3a) Richard Stewart, Duke of Rothesay (b.1496: d.1502)
4a) Claude Stewart, Princess of England (b.1499)
5a) James V of Scotland (b.1502)
6a) Alexander Stewart, Duke of Ross (b.1503: d.1507)
7a) Mary Stewart, Princess of England (b.1506: d.1506)
3a) Richard Plantagenet, 1st Duke of York (b.1478: d.1542) m. Germaine of Foix (b.1488: d.1530) (a)
1a) Edmund Plantagenet, 2nd Duke of York (b.1509)
2a) Elizabeth Plantagenet of York (b.1517)
4a) Anne Plantagenet, Princess of England (b.1481: d.1481)
5a) Margaret Plantagenet, Princess of England (b.1483: d.1498) m. Maximilian I, Holy Roman Emperor (b.1459: d.1519) (a)
1a) Stillborn Son (c.1498)