Polygamy (as in being married to more than one person) isn't and wasn't legal in Japan. I don't see how it would in any way be feasible or acceptable to take a new wife in Japan of all places, where honor and family are such important aspects of their culture, while already married.
I understand OP's question to be more practical than honour and family - literally as "Would the Japanese recognise a marriage performed in England?".
Civil registration started in England only in 1837. Modern koseki system in Japan started in 1872, but different systems of civil registration existed way before that, including one created under the shogunate (though many years after Adams's passing).
And indeed, they couldn't possibly careless about his married status regarding him marrying a Japanese woman. His declaration of death was so that he would be free to serve the shogunate freely, without the obligation of return to his family in England.
