...Yes, many Normans went to Wales, displacing the natives. For five+ centuries they used patronymics so names are not as useful there. There might be some name projects that would be helpful. I looked up the HG *a1b and British Isles and there are only 200, seven that mention Wales. So few get the deep clade testing done.
I don't think the Normans sent to Wales displaced the natives. They were quite adept at leveraging and intermarrying with the native Welsh. I've read the "Normans in South Wales" and my perspective is they were more of a case of ruling elite than massive invasion.