There is no upgrade path, they are completely separate plugins. I would suggest you continue using mod_certificate for your existing certificates (and making any code changes to the plugin if necessary when you upgrade) and then use mod_customcert for new certificates (or if you are more comfortable with mod_certificate continue using that).
The reason there is no upgrade path can be read at https://moodle.org/mod/forum/discuss.php?d=229683.
Thanks for providing the information. Now I see, the term 'upgrade' is not applicable to the certificates. What I meant was, we want to upgrade from Moodle 3.1 to 3.5. According to https://moodle.org/plugins/pluginversions.php?plugin=mod_certificate there is no mod_certificate version for Moodle 3.5. That is our problem.
My guess: I deactivate mod_certificate in 3.1, upgrade the site to 3.5 then install mod_customcert. The participants who already have a mod_certificate (or more) will continue to have it (them) after the Moodle upgrade. Any new certificates issued after the upgrade will be of type mod_customcert. So old users may get a mixture of mod_certificates and mod_customcerts. Am I right?
Or, I can think of installing mod_custermcert on the current Moodle 3.1 in parallel with mod_certificate. Then deactivate mod_certificate before upgrading to Moodle 3.5 - upgrading the mod_customcert at the same time.
It is important that the users do not lose mod_certificates.
You can keep mod_certificate as long as you want, and it may very well work with 3.5 with no or minimal effort. There is no reason to abandon it if you are happy with how that one works. Have you tried using the version of that certificate on Moodle 3.5? If you install mod_customcert you are going to have to redesign the certificate which won't be necessary if the old version works perfectly fine on 3.5.
Looks like I misunderstood that mod_certificate will not run in Moodle 3.5. So I will go ahead and upgrade Moodle and then see whether we need mod_customcert. Will report back, please have patience, it is the summer break.