When I saw the movie 'Conquest of Paradise', which depicts Columbus arriving in America in 1492, I saw there were soldiers using hallberds and arquebuses and matchlock muskets. Some historians argue that this period can still be called the (late) middle ages, so I asked myself, how integrated were black powder based weapons by then? Had they already become standard or did many European armies still use bows and crossbows?
There does some to be some reason to believe some of Sir Walter Ralegh's may have been armed with bows and crossbows. Here's an article on the National Park Service website that discusses it succincly. The main source for this article is, The Arms and Armor of Raleigh's Roanoke Voyages, 1584-1590, by Phillip W. Evans. The article gives accounts of two advice pieces from the 1580's (one by Richard Hakluyt and the other is anonymous) written to advise Ralegh on how he should carry out his exposition. They both state that Ralegh should bring along archers. Whether he actually did or not, seems to be a matter of debate as the actual primary sources we have Roanoke don't mention them specifically.
In addition, there seems to be archaeological/historical evidence for Spanish use of crossbows in the Americas. Bolt/arrowheads have been found at Spanish camps/settlements in Florida and there are accounts of at least some of de Soto and Cortez's men being armed with crossbows.
Regardless, crossbows and longbows still had a couple of advantages over firearms despite largely falling out of favor by this point, especially in a frontier/wilderness environment.
*Arrows/bolts were reusable if not broken
*bows at least were far quieter
*they did not produce smoke
*they could be used in wet conditions (though this was still not advisable)
*early firearms such as arquebuses and matchlock muskets were relatively cumbersome to use
These factors alone were enough to ensure that bows and crossbows still held a place in the hands of early colonists. As the article states, however, it is unclear whether these weapons were actually ever used in conflicts with Natives at Roanoke.