'Mutually' is definitely wrong. If you said that to someone you've just met, they would be confused. The word exists, of course - I'm guessing that you found it in a dictionary when you were looking for a translation from an equivalent word in Russian - but that isn't how we use it in English. 'Mutually' is an ordinary adverb which has to be part of a sentence, for example, "The two diplomats negotiated a solution which was mutually acceptable." It cannpt stand alone in a conversation.
The word you were trying to find is 'Likewise'. If someone says something to you and you want to say "Yes, it's the same for me" or "Yes, I think that, too", you can respond by simply saying 'Likewise'. It's a little short, which can come over as cold or casual, depending on the circumstances. But if you say it in a genuine way, with a smile and some eye contact, it's perfectly polite.
Otherwise you can say either 'Nice to meet YOU' (with the emphasis on the 'you') or 'Nice to meet you, too'.
I hope that helps.