You're right. There's a respectably wide span across which the longbow may have become dominant, and another period of using some amount of both. I think most certainty is their origination in Wales. The Welsh and Normans were constantly learning from each other's tactics from the earliest excursions by the Normans. There are some who document longbows being used during the war of succession between England and Normandy in the 1140s (The Anarchy), and in Wales a little later in the 12th.

Others have argued much later. Unless we get an older find of Mary Rose quality, I think that debate will run on. :)