Google announced the move in a blog post, saying their analysis showed that most sites shown in search results were “good to go” for mobile-first indexing. It said 70% of those in the results had already been moved across. Between now and September, Google will continue moving sites across when ready.
“To simplify, we'll be switching to mobile-first indexing for all websites starting September 2020. In the meantime, we'll continue moving sites to mobile-first indexing when our systems recognize that they're ready.”
Google began the first mobile-first indexing tests in 2016, with the first wider rollout two years ago.
You can check in Search Console to see if your site has been migrated over. The status can be seen in Settings or in the URL Inspection Tool. You'll see whether your site was crawled Googlebot Desktop or Googlebot smartphone.
Google’s John Mueller confirmed on Twitter that the move applies to all websites globally.
If your site hasn’t yet been migrated across, the time to get ready for mobile-first indexing is now.
Google has extensive support and best practice for websites to get into the best shape for this here.