Is a vow to fast for two months considered valid if it was not uttered initially, but rather communicated later in the past tense? And is expiation required for it?

What transpired from you is not considered a vow (nadhr), because a vow is not established by mere intention. It requires a verbal utterance that indicates commitment. As for what your mother told you, that you had made a vow, it appears to be merely an incorrect report, and therefore, nothing is binding therein.