Since Wales was invaded and the English tried to incorporate it, at what point did it start to be recognised as a country rather than part of the United Kingdom?
Wales is part of the United Kingdom, and was part of the UK's predecessor states Great Britain and before that England.
Wales' status as a nation within the state was more a cultural matter than a legal or political one: before contemporary devolution (discussion of which in depth is precluded by the 20 year rule), Wales was integrated into England for nearly every important legal and political purpose -- a strong contrast with Scotland and Ireland which had considerable legal and political distinction even before contemporary devolution.