I would define a mature security process not by the process itself, but by the results (essentially the SLA's we expect), some specific things that quickly come to mind:
1) Ability to contact the vendor.project/individual (securely) to report flaws, at a minimum an email address, ideally also a PGP encryption key, even better a way to securely file security bugs into the bug tracker (but increasingly not due to widespread GitHub usage)
2) Acknowledgement and follow up of security reports, I would say max a week to have a human reply to a legitimate issue/question. For example I've had the Puppet security team reply within 3 minutes on a Friday afternoon. At Red Hat we have an SLA for human replies (not fix mind you, but a reply at least) for security reports sent to us.
3) Bug tracking - projects have to have a bug tracking system
4) Projects actually deal with and address the security flaws/reports, even if it's a "WONTFIX because X/Y/Z" or "We'll get that in the next release in 6 months", but there needs to be some sort of resolution. Having bugs sit in a bug tracker for years does nobody any favors, but at least if they're labeled correctly the users of the project can assess the risk/do workarounds/etc.
5) Projects have some notification process for security updates, minimally changelogs (ideally with CVEs), even better a security web page, even better an announcement list (email or other)
But the above really all boils down to: what do they do with security vulns/reports? The more they do to get them addressed/fixed, the more likely they are to be ready to be a CNA, and more importantly, the more likely them being a CNA will have some value. It's simply a question of "does this add value", e.g. a project with a less mature response process, but becomes a CNA and faithfully and correctly assigns CVEs to all the security vulnerabilities found, even if they don't fix them, would still add value in the sense that now I can assess that projects security, or perhaps start helping out with code commits to fix those CVEs. Obviously I'd prefer to have CNA's that assign and fix CVEs, but we all have to start somewhere. Labeling our security vulnerabilities properly as such is a good start (then we at least know what needs to be fixed).