Influence of different reference genomes used for mapping

I am trying to integrate 2 scRNAseq datasets: one previously published where the authors used the refdata-cellranger-mm10-1.2.0 reference in Cell Ranger, and one I have generated where I used the refdata-gex-mm10-2020-A reference in Cell Ranger.

I was wondering what would be the better approach to tackle this problem. Is it important for the integration process? How should I approach the analysis? Is it possible to convert their gene names to mines? Do I have to re-align my reads to the same reference files used by the other lab?

Looking forward to your ideas!