When the new feature becomes available, tap the paperclip icon when you're composing or replying to a message in the Gmail app. This will bring up a sheet with carousels for "Camera roll" and "Attachments" (formerly named "Recent attachments").
This new Attachments section will feature a special folder icon for directly accessing the contents of the Files app on your iPhone or iPad, including access to the native search and selection tools of the Files extension.
As for the rollout timeline for the Files integration, Google says there will be an extended rollout which may take longer than two weeks for "feature visibility." Whether that means this is a server-side change or requires an updated version of the app is unclear.
The Gmail app is a free download available on the App Store. [Direct Link]