I can offer a partial answer to this question but I readily defer to other historians for whom this area is their primary expertise.

Henry VII was originally unwilling to support an expedition west in search of a new sea route to markets in the east. He had met with Christopher Columbus' younger brother Bartolomeo in 1489 and had apparently been unimpressed with his condition and proposal. Long after the fact, Sir Francis Bacon recorded in the early 1620s that Bartolomeo had been captured by pirates on his journey to England and had arrived at court in a state that did not well serve his cause. "And it so fortuned that he was taken by pirates at sea, by which accidental implement he was long here he came to the King; so long that before he had received a capitulation with the King for his brother the enterprise by him was achieved, and so the West-Indies by providence were then reserved for the crown of Castilia." Sir Francis Bacon, The History of the Reign of Henry VII (Cambridge: CUP, 1998), 157

There does appear to be evidence that English ships had nonetheless been searching west even before the first successful voyage of Christopher. The Spanish envoy Pedro de Ayala in a letter to his king stated in 1498, "Merchants of Bristol have for the last seven years annually sent out ships in search of the island of Brazil and the seven cities." Bergenroth, Calendar I. 177, no. 210.

Following Christopher's first successful voyage to the New World in 1492 and his safe return the following year Henry VII changed his mind regarding the potential possibilities for trade via the west and chose to give his support to the Genoese explorer John Cabot. In exchange for a trading monopoly to all lands discovered he would license Cabot and five ships to undertake such an expedition. However it is important to note that the King did not financially support Cabot in this endeavour. The King issued him letters patent in 1496 but it was left to Cabot and his partners to finance the expedition themselves. It appears that Cabot decided to assign a portion of his rights by charter to his backers and was effectively "mortgaging his future." Evan Jones, "The Matthew of Bristol and the Financiers of John Cabot's 1497 Voyage to North America" English Historical Review Vol. 121, No. 492 (Jun., 2006), pp. 781

Thankfully for Cabot he successfully crossed the Atlantic for the first time in 1497 under English banners. He returned that year for an audience with the King and was celebrated at court for his achievements. He received a substantial sum and was assigned a pension to support him in future endeavours. For decades thereafter Bristol would become a centre for financing and supporting expeditions of this kind to the the New World in pursuit of new routes to Orient. David Harris Sacks, The Widening Gate: Bristol and the Atlantic Economy, 1450-1700 (Berkeley: University of California Press, 1991), 51.

The King would continue to support other expeditions to the west from Bristol until the end of his reign.