If you are starting with NCBI GeneIDs, you can use the gene_orthologs.gz file located on the NCBI FTP site. It has data in a tab-delimited file with the following fields:

     #tax_id [  1]: 9606
      GeneID [  2]: 1
relationship [  3]: Ortholog
Other_tax_id [  4]: 10090
Other_GeneID [  5]: 117586

This table has data for organisms other than human and mouse. You can filter only the human:mouse rows using awk as shown below:

zcat gene_orthologs.gz | awk 'BEGIN{FS="\t";OFS="\t"}($1~/^#/||($1==9606 && $4==10090))' > hs_mm_orthologs.tsv

Try biobtreeR with its orthologs support for these type of conversions. Followings are examples for converting gene names,ensembl and NCBI identifiers, similarly vice versa can be done.

ensembl identifiers to human identifers and gene names

bbMapping('ENSMUSG00000059552,ENSMUSG00000023456','map(ortholog).filter(ensembl.genome=="homo_sapiens")',attrs = "name")

gene names to human identifers and gene names

bbMapping('tpi1,shh','filter(ensembl.genome=="mus_musculus").map(ortholog).filter(ensembl.genome=="homo_sapiens")',attrs = "name")

NCBI mouse identifiers to human identifers

bbMapping('22059,21991','map(ensembl).map(ortholog).filter(ensembl.genome=="homo_sapiens").map(entrez)')

Try this https://www.r-bloggers.com/2016/10/converting-mouse-to-human-gene-names-with-biomart-package/

or this https://syngoportal.org/convert.html