I'm not sure that it would make a difference. As a single taxpayer, your fiancee can deduct student loan interest (if they are paying any). You could also deduct the interest if married filing jointly.
The difference in tax owed between two people filing single and one couple filing jointly is complex. It used to be that taxes were often higher for married couples (the so-called "marriage penalty") and there have been attempts to fix it in the tax code, so I can't really say, without testing the exact circumstances, whether joint or single filing will be better overall.
If you are married on 12/31/2020, you will file as married for the whole year, so "early" only means anything if you are talking about 2021 or 2022 vs 2020.
