It would seem that Google is just rounding off the global release of the new Search Console. Most people in the world should now have access.

To check if your access has rolled out, go to https://search.google.com/search-console and check the top right of the page.  If you can click on the “search properties” button in the top right to see all of your verifies sites in the new console.

For those of you that are not particularly fond of change, you can still access the old console (as well as a link to the new) using the link at the bottom left of the interface.