I understood that all clubs were ordered by Franco to get rid of English spelling of names so Athletic Madrid became Atleti and Barcelona FC became FC Barcelona. Only Athletic Bilbao told him to do one!
Athletic were indeed known as Atletico Bilbao under Franco but changed back after he died.
Athletic Madrid (or, to be accurate, Athletic Aviacion de Madrid) became Atletico Aviacion de Madrid in the early 40s and then simply Club Atletico de Madrid in the late 40s, which they have remained ever since.