There was a branch of the Yorkist family tree that survived Bosworth and the events that followed. The sons of Elizabeth of York, that is the new Queen's great aunt, survived.
One, John, supported the pretender Lambert Simnel, but came to a sticky end in the Battle of Stoke, 1487. Another, Edmund, challengingly known as 'The White Rose', fled to the Low countries where he was welcomed by the Burgundy court. He was handed over to Henry in the early 1500s as part of an Anglo-Burgundy peace treaty, and promptly banged up in the Tower of London, later to be executed by Henry VIII. Richard, the youngest of the sons, stayed abroad, but was killed at Pavia in 1525.
These three brothers must have been a real worry for the King, especially as they had support from the influential Burgundy court. They must have been like loose cannons to Henry's security.
We now move on to the two pretenders. Both of the attempts by Lambert Simnel and the Perkin Warbeck seem like fanciful tales, rather than historical truth. You couldn't make up more bizarre stories about attempts to usurp the King. But we must be careful not to be too flippant about these two uprisings, as they must have alarmed Henry considerably.
Lambert Simnel was a dead ringer for the Earl of Warwick, and so he claimed to be this grandson of Edward IV. He secured the support of Margaret of Burgundy (always a supporter of Yorkist claimants) and John de la Pole. He made his way with 2000 mercenaries to Ireland, where he was crowned King of England, and then to England. He landed on the Lancashire coast, where the families that had supported Richard III and had suffered hardship under Henry met him. To Henry, the threat was very real, he had no way of assessing the amount of support Simnel had mustered on his route south. These forces met the King's army at Stoke on 24th May 1487, where they suffered a complete defeat. But it wasn't until 1497 that Simnel was handed over to Henry.
There was a rather large flaw in the plans of Lambert Simnel. The real Earl of Warwick had been in Henry's custody since he became King. There was no way that he could have been Edward Warwick. He was in fact the son of a joiner from Oxford! Henry considered him harmless and, instead of being executed, he was allowed to live out his life in the royal kitchens.
Like Lambert Simnel, Perkin Warbeck claimed to be an heir to the throne of England. He was a Flemish man, and like Simnel had the backing of a foreign country. This time it was France. The French King, Charles VIII was eager to distract Henry away from his designs on Brittany. In addition, Warbeck also secured Scottish aid. James IV welcomed Warbeck in 1495, marrying him to a rich Scottish aristocrat.
However, in 1497, Henry renewed a truce with Scotland and James was forced to abandon his Flemish pretender. Warbeck fled to the south of England, and was arrested while trying to escape from the port of Southampton.
Initially, Henry showed clemency. But he abused this clemency and was executed in November 1499.
This saw the last of the pretenders, and from this point onwards Henry ruled in relative security.
John Guy says that this was Henry's major threat, and so it was. This time the threat was not of a dynastic nature, it was a tax riot that turned into full-scale rebellion.
The tax riot began in Cornwall in 1497. The Cornishmen felt that it was unfair that they should be taxed to fund a campaign in Scotland, and that by tradition the northern counties should bear the brunt of this taxation.
The riot quickly grew into a rebellion as 15 000 rebels marched to Exeter, Salisbury, Winchester, and then on to Kent.
The threat to the capital was so severe that London was called to arms. On the 17th June, the rebels faced the King's forces at Blackheath - only a few miles away from London, and dangerously close to the royal arsenal at Greenwich. Thousands were slain and the three ringleaders, Michael Joseph, James Touchet and Lord Audley were captured, taken to the tower, tried and executed. Their heads were raised on poles on London Bridge as a warning to others who considered insurrection as a form of protest.
If you are asked a question on the threats to Henry's reign you should consider threats from abroad, and these 4 other threats.
The Cornish rebellion must have been thoroughly alarming for the citizens of London, but we need to remind ourselves of what the motives of these rebels were. They were driven by a sense of unfairness and by economic concerns. They did not intend to usurp the crown, and they did not have the backing of another European monarch. What do you suppose these Cornishmen would have done if they had reached Henry VII, and been able to make demands on him? Henry may have lost face, and been forced to retract the tax, had the rebels had succeeded, but I doubt that it would have led to him losing his crown.
The other threats were direct threats to the throne of England. In the cases of Lambert Simnel and Perkin Warbeck, they were actual challenges to the throne. In hindsight these attempts seem laughable - that a joiner's son should be crowned King by the Irish Lords, and that a Flemish man should be accepted as the rightful heir to England at the Scottish court. The reason why Henry took these attempts so seriously at the time, and by historians, is due to the level of encouragement and actual support these pretenders got from foreign princes. In fact it is widely accepted that these two were pawns for European powers.
But, the largest potential threat to Henry's security came from the de la Poles in exile. They did not only threaten Henry VII's stability, but Henry VIII's also. Even at Henry VII's death in 1509 there was a Yorkist champion alive and able at any time to act as a figurehead for a Yorkist uprising.