Which is more common to say in the U.K. or in the U.S.,"Japanese government is going to adopt English as an official second language, or a second official language?"
It depends. I would assume that it would be an 'official second language', meaning that it would be the foreign language taught in schools. If Japan adopted English as a 'second official language', then English would be a language of the state, alongside Japanese, which seems less likely. ;-)