In the UniProtKB page there is a tool to make that conversion for you. I can see the KEGG is in their list...You can also use BioMart Central: under 'Interactions and pathways' click on Reactome and under filters choose KEGG IDs. If you are not too familiar with BioMart, it is a data retrieval tool (online version but programmatic access too) that goes through four steps: data, filters, attributes and results. There is an youtube tutorial video showing how to go through those steps using the Ensembl BioMart. It should be similar when using Reactome BioMart.

1. KEGG mapper convert ID http://www.genome.jp/kegg/tool/conv_id.html

2. uniprot Retrieve/ID mapping www.uniprot.org/uploadlists

using clusterProfiler:

https://guangchuangyu.github.io/2016/05/convert-biological-id-with-kegg-api-using-clusterprofiler/

If you want to do this locally, you can download idmapping file form Uniprot ftp://ftp.uniprot.org/pub/databases/uniprot/current_release/knowledgebase/idmapping/. Then with some simple awk/grep unix commands or a couple of python/perl scripts, you will be able to extract uniprot id(s) of the correspondent kegg id and vice versa.

DAVID (Database for Annotation, Visualization and Integrated Discovery) has a Gene ID Conversion Tool: https://david.ncifcrf.gov/conversion.jsp