While we have Warkworth mentioning "three hundred of Flemmynges with handegonnes" leading the way for Edward IV, but no mention of crossbows, that may just be due to the novelty value.
Crossbows would have been quite common, even amongst the English - they had to be specifically prohibited as an alternative to longbows for practice under Henry VII (or VIII, I forget), whereas they were previously acceptable.
French, especially Gascons, were likely to be crossbowmen.
Either would not look out of place in the WotR, even the Pastons hired a handgunner to defend the manor.