She was his wife all along. It's just that the word "wife" is used only when the woman is being referred to in relation to the man.
In Hebrew the difference between "woman" and "wife" is just a matter of whether there is a possessive suffix. In English we don’t say, "her man" or "his woman" (except with certain connotations). Instead we change the words to "husband" and "wife" when we add a possessive modifier.
In Hebrew (at least in the Genesis 2-3),ʾiššâ is usually translated "woman", but ʾištô (literally "his woman") is usually translated "his wife". In Genesis 2:25, shortly after God brought the woman to Adam, she is referred to as his woman, i.e. (as in the vast majority of translations), his wife.
וַיִּבֶן֩ יְהוָ֨ה אֱלֹהִ֧ים ׀ אֶֽת־הַצֵּלָ֛ע אֲשֶׁר־לָקַ֥ח מִן־הָֽאָדָ֖ם
לְאִשָּׁ֑ה וַיְבִאֶ֖הָ אֶל־הָֽאָדָֽם׃ (BHS)
wayyiben ʾădōnāy ʾĕlōhîm ʾet–haṣṣēlāʿ ʾăšer–lāqaḥ min–hāʾādām ləʾiššâ waybiʾehā ʾel–hāʾādām
And the rib
that the LORD God had taken from the man he made into a woman and
brought her to the man. (ESV)
וַיִּֽהְי֤וּ שְׁנֵיהֶם֙ עֲרוּמִּ֔ים הָֽאָדָ֖ם וְאִשְׁתּ֑וֹ וְלֹ֖א יִתְבֹּשָֽׁשׁוּ׃ (BHS)
wayyihyû šənêhem ʿărûmmîm hāʾādām wəʾištô wəlōʾ yitbōšāšû
And the man and his wife were both naked and were not ashamed. (ESV)