Of course it’s officially called the “British Army” - but you will find it generally called the “English Army” on the continent, and such usage crops up everywhere in reports by ordinary civilians of eg. events in the second world war. Furthermore, whether Wales is altogether distinct from England is not so clear cut as you suggest. It is only a principality, not a kingdom. When Elizabeth I or Henry VIII declared “England is an empire”, meaning a sovereign state, they included Wales. England and Wales together still form the constitutional successor to the former Kingdom of England. Most tellingly, Wales follows the same legal system - including not just the English ‘common law’, but the machinery of justice also.
Scotland of course emphatically does not, and yet to the ire of nationalists Scottish troops are still frequently thought of worldwide as forming part of the ‘English’ army! This must be galling, but it’s still true, and is likely to remain so for some time.
