The civil war only really affected the nobility.
A war that deeply affect the elites of a country and its structures certainly had an impact over the said country.
The cost of this conflict (for example, mercenaries but not only) did impacted royal and nobility treasury : there's reasons why Henry VII was hoarding wealth, and one of these was to compensate what was wasted on war effort and the bankruptcy caused by Woodville.
Over-going a colonialism effort when you're trying to build a strong royal treasury back is certainly something you may spare if the results simply don't look promising enough.
.He's rich,but too much of a Miser.
That's really an exaggeration. While his personality may have influenced, the most important point is that when he get the throne, he get a huge problem of treasury, namely an empty one.
Of course, renewed fiscal effort wasn't exactly popular, especially with the growingly important merchant class or the nobility that wasn't much used to pay fully its taxes, but it was eventually as much invested than spared, with (for example) development of Portsmouth.
That said, I could see him funding Colombus as he funded Cabot, if he thought there was something to gain for. But giving he didn't fund Cabot until Colombus found something, it would ask quite something to convince him, and I'm not too sure discovering Basque fishermen and cod hoarding will met much enthusiasm with him.
Nevertheless, it could be a first step for English semi-permanent settlement in NewFoundland and a greater motivation for Castille/Aragon to look on the western exploration.
As for Iberian powers,the English will have some sort of head start and the English would probably have the support of the pope in eradicating Iberian New World settlements under the view that they were 'illegal' as the Spanish did the English ones in real life.
You shouldn't overlook the religious part of destroying these establishments. They did the same with French protestants settlement of Florida, for instance, as much because they were within their sphere of influence that because they represented an heretic religious presence.
The pontifical support would certainly not play, at least immediately and as long Spain doesn't turn Protestant (which may certainly take a while, to say the least).
The problem is not having an head start, but a question of naval dominance : Iberian kingdoms were truly maritime powers and able to kick others out of their sphere if needed in the late XVth century.
I would also point trying to be at odds with Castille/Aragon and Portugal would be quite contradictory with the diplomatic activity of Henry VII that was more about getting closer of Trastamares. He would certainly search to get a compromise as much he could, would it be only to have free hands elsewhere.