Your subscription moves with your Apple ID. Your blocks, folders and App Limits don't, so you'll set those up again on the new phone.
What comes with you
Your subscription, as long as you use the same Apple ID.
Your free week. It doesn't start again on a new phone.
What you'll set up again
Quick Blocks, Focus Sessions and App Limits
Folders, and the apps in them
Apple ties the apps you pick to the phone you picked them on, for privacy. Your blocks are also stored on the phone, not in iCloud, so they can't be copied across.
On your new iPhone
Install Slow Time Club and open it.
Allow Screen Time access when asked.
If you're asked whether to restore from iCloud, tap Restore from iCloud to bring back a few settings, such as your name and break length.
Slow Time Club finds your subscription by itself. If you see the plans screen, tap Restore.
Create your blocks and folders again, and pick your apps.
Tip: before you switch, take a screenshot of your Focus Sessions and App Limits on the old phone so they're quick to recreate.