When you sign up for a Floranext website, for as long as your Floranext site is in "preview mode" (ie. not a live website at your domain), your current website will still be accessible at your current web address as normal—nothing will change from the perspective of your customers until you are ready to make the switch.
When the Floranext website is launched, it will replace the existing website at your domain. The original website may still be accessible in an editor through your old website service, though it won't be live anymore.
There will be no downtime between your old website coming down and your Floranext website being launched.