I know this post is 14 days old, but I noticed it doesn't have an answer and I'm somewhat knowledgeable on this topic:

I've read some of that book too (for a research project) and what Jones says is right, however, it's important to note that not all Plantagenets were equal in threat or royal lineage. For example, the De la Pole family was the primary Plantagenet threat during the Tudor dynasty. Most of the threat was eliminated during Henry VII's reign, and Henry VII faced little threat from them overall.

Some of the threat was eliminated fairly peacefully, as not all of the De la Poles were keen on overthrowing the King. John De la Pole, 2nd Duke of Suffolk, for example, didn't fight against Henry at Bosworth and ended up participating in his early parliaments as well as administering his will throughout much of the country during the late 1480s. He died naturally.

Others of the family, such as John De la Pole (the Earl of Lincoln), did fight at Bosworth. John actually survived the battle and was later killed, in 1487, at the Battle of Stoke which is widely regarded as the last battle in the Wars of the Roses, representing the end of any significant Plantagenet threat to the Tudor dynasty. A year prior, Sir Humphrey Stafford and Francis Lovell, 1st Viscount Lovell (and one of Richard III's closest allies & friends) tried to launch the Stafford and Lovell Rebellion but failed due to Henry VII's use of spies an informers - Stafford was executed and Lovell fled, never heard from again.

Importantly, the most credible Plantagenet heir to the throne - Edward Plantagenet, 17th Earl of Warwick - was still alive at this point. There was little chance he'd usurp the crown so he was left alive until 1499 when he was executed alongside Perkin Warbeck (one of the two 'pretenders' of Henry's reign, the other being Lambert Simnel, who was not executed, who are both also Plantagenet threats that Henry eliminated, however, neither were actually descendants).

To my knowledge, the last threat was Edmund De la Pole, 3rd Duke of Suffolk, who was in exile in France - in Aix from 1502 to 1504 - and only returned to England in 1505/6 after Emperor Maximilian I was shipwrecked and at Henry VII's mercy, eventually agreeing to return the Duke to England in exchange for his own safe passage. He was executed in 1513 under Henry VIII.

I may have missed some details out during Henry VIII's reign but I think that's a summary that is satisfactory, I think.