I thought this was going to take hours, but it took just a few minutes! There were some hints that it was related to namespace::clean or namespace::autoclean, so I checked the Moose install output for "clean". I found only the following:

[...]
  CPAN.pm: Going to build D/DO/DOY/Package-Stash-0.23.tar.gz

***
    Conflicts detected for Package::Stash:
      MooseX::Role::WithOverloading is version 0.08, but must be great
+er than version 0.08
      namespace::clean is version 0.18, but must be greater than versi
+on 0.18
***
Checking if your kit is complete...
Looks good
[...]
[download]

There were no corresponding installation of those dependencies. Upgrading namespace::clean fixed the installed version of MooseX-Types that got broken, and allowed me to install 0.25 cleanly.

Wow, that did it! I upgraded namespace-clean from 0.18 to 0.20 and everything works again.

That did it! I updated namespace::clean from 0.18 to 0.20 and everything is fine again!

Thanks for your help!