Any mentioning of English Longbows used in battle that I have encountered have been against the French, Irish and Scotts. I cannot recall any accounts of English Longbows (or even archers) during the Crusades, while I see no reason to think they were simply not present. Such accounts might be present, as I am not a professional historical researcher myself, but their fame has only been against the Scotts and French, which werent well known for their archery.

No I think I will rather go for archers in the middle east of Arab, Iranian, Armenian, Greek or Turkish origin, regardless whether they fought for Islam or the Byzantines, both mounted as well as on foot.