I'm honestly not certain about bows specifically - it's entirely possible that John Cabot brought longbows with him on his expeditions. The longbow wasn't fully withdrawn from English armies until the 1590s, so Walter Raleigh could have brought them with him to North America, but I have to confess that it's a little beyond my usual area.

What I can provide an answer about is the use of crossbows, particularly by Spanish conquistadors in Central America. The short answer is that yes, crossbows very much were a part of the standard armament of early conquistador armies.

Bernal Diaz wrote in his account of the Spanish-Aztec wars that when Cortés initial expedition of about 600 set out it had 33 crossbowmen and 13 arquebusiers. Later, when describing the La Noche Triste, the forced flight of the conquistadors as they were driven from Tenochtitlan, he gives the enlarged Spanish forces of1300 soldiers including 80 crossbowmen and "a like number of musketeers". Many Spanish soldiers died fleeing Tenochtitlan but were quickly replaced so that when they returned and laid siege to the city they had 118 crossbowmen with them. So crossbowmen certainly were present with Hernan Cortés.

Now, a few caveats. Bernal Diaz was an eyewitness participant during these wars, but he was also writing his account some 50 years after the events and had a bit of an axe to grind (whether that was a fair position or not is a separate discussion). That means we should take these figures with a huge grain of salt. It should also be emphasised that Spanish soldiers made up a small fraction of the overall armies that fought. The crossbowmen were already a small proportion of the Spanish army and they in turn were a small part of the overall army so we can't overstate how important the crossbow was to the eventual fall of Tenochtitlan and the collapse of the Triple Alliance.

Diaz isn't our only evidence, though. Inga Clenninden has made the interesting point that recruitment agreements for crossbowmen on these expeditions give them an equal share of any loot acquired as an arquebusier would receive - strongly suggesting that these were seen as soldiers of comparable value.

The use of the crossbow didn't end with the fall of Tenochtitlan in 1521. Francisco Pizarro brought crossbows with him to Peru, although there is far less detail about how they were used. Francisco de Orellana brought crossbows with him in 1541 when he joined Pizarro's half brother Gonzalo on the search for the "Land of Cinnamon". Orellana would eventually get stuck on the Amazon river and have to travel the full length of the river, emerging into the Atlantic months later. Friar Gaspar de Carjaval, who traveled with Orellana and later wrote an account of what he'd seen, credits the crossbow with being essential for their survival.

The real issue with using guns in the Americas wasn't the guns themselves, or even the ammunition, it was the gunpowder. Manufacturing saltpeter in particular was challenging and required the right environment to succeed - this meant that basically all of it was made in Europe. A gun without gunpowder is pretty useless, and transporting steady streams of the stuff to the Americas was expensive. Guns were also less effective in damp climates, like rain forests, and their main advantage wasn't as important. Guns are far more powerful than bows/crossbows but when fighting more lightly armoured opponents that difference matters less. Now, the Americas were not without armour and the padded armour used in some regions could be relatively effective at stopping light projectiles so the advantage of guns wasn't entirely absent, it was just a bit diminished.

As to when the crossbow stopped being used in the Americas, I honestly don't have an answer. I tried tracking this down for my book and I got as late as Orellana but I'm not a specialist in colonial history and I didn't have the time to become one so I've tabled it for a future project. From the bit of research I've done my expectation is that it largely fell out of use over the course of the sixteenth century but I could see it potentially lasting into the early seventeenth century. I'm hoping to pick this project up again in future so hopefully in a few years time I'll be able to offer a more concrete date!

Not to toot my own horn too much, but I wrote a whole section about this in my own book: The Medieval Crossbow by Stuart Ellis-Gorman

Other sources include but are not limited to:

Matthew Restall, When Montezuma Met Cortés

Inga Clendinnen ‘Fierce and Unnatural Cruelty’, in The Cost of Courage in Aztec Society: Essays on Mesoamerican Society and Culture

Ross Hassig, Mexico and the Spanish Conquest,