The full answer
The lock lives on the server, so signing in again on the same account brings back your protection, the day count unbroken, your keyholder, and the deletion delay as you set it, and a reinstall can never bring back a weaker one. Three things do have to be done again on the phone: granting Screen Time, picking your apps and turning site blocking on, because Apple never lets the app pick leave the phone it was made on and site blocking is a profile installed on the phone itself. Until they are done the shield is not applied, and Fold’em says so instead of claiming to be protecting you. Until Screen Time is allowed again, your keyholder and supporters see Protection off. Once it is, they see Not confirmed yet until your apps are picked again, and Protection off again if that takes more than 12 hours.
The lock is not stored in the app
Fold’em keeps the record of your protection on the server, not in the app's own files, and that is the whole reason deleting the app achieves nothing. The record that says protection is running, who holds the key and whether the deletion delay was on outlives the install. Sign in again with the same account and the app reads it back and puts the block up again.
If you turned the deletion delay on, this question does not arise for a while: while that restriction holds, no app on the phone can be deleted at all, and taking it off starts its own 72 hour wait on the server.
The three things that are done again
Screen Time permission is granted per install, so a new install asks for it again, and site blocking is a profile the phone itself holds, so a fresh install turns it on again from the Sites page on Lock. The app pick is the same story for a stricter reason: the picker hands Fold’em opaque tokens that are tied to the phone and the install that made them, and Fold’em never uploads them. Only the count of what you blocked goes to the server, which is exactly why nobody else can ever learn which apps you blocked.
So the pick is redone by hand, and so are any sites you added, which came from the same picker; there is no version of Fold’em that could do it for you. Everything else comes back untouched: your protection, the day count with no break in it, your keyholder, and the deletion delay at its old setting. Money kept is worked out from your weekly spend, which is kept on the phone, so it reads zero until you enter it again under Weekly gambling spend on the Money kept page.
The app reports what is outstanding on every reading. Until Screen Time is allowed on the fresh install, your keyholder and supporters see Protection off, because a phone without the permission is a phone with the shield off.
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Allow Screen Time again
After signing in, Today shows a card that reads Apps need attention, or 2 layers need attention while site blocking is off too, with the line Screen Time is off. Tap it and the Apps page on the Lock tab says what is missing: Screen Time is off, so gambling apps are open, with one button, Allow Screen Time, which raises Apple's own request. Nothing here is a new setup: it is the same protection being put back on its feet, and SOS, People and Settings stay open while you do it.
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Pick the apps again
With Screen Time allowed, the same sheet reads Your app choices need picking again, so gambling apps are open, and Choose apps opens Apple's own picker. Pick everything you had, and anything you have thought of since: adding is always allowed, and a pick holds for the rest of your protection.

The count, back at zero
Why the block does not drift after that
On every launch and every time you bring the app to the foreground, Fold’em checks two things: that Screen Time is still granted, and that the settings it applied are still on the phone. If anything has drifted while the record says protection is running, it applies the block again without asking and reports what it found in the same reading.
That repair puts the block back the way your record describes it, and it has no path to ending protection: that is written by the server, and the phone applies it. A reinstall, a restore from a backup and a phone that was off for a week all end in the same place, which is the block as the server describes it.
What this does not do
It does not carry the app pick across for you, and it never will, because the tokens that name your apps are kept on the phone by Apple and are not Fold’em’s to move.
It does not reach a second device. Signing in on another iPhone puts the record there, but the block on any phone is only as good as the permission and the pick on that phone.
And it is not a way to change your mind. Reinstalling does not end protection, does not start a request to end protection, does not shorten a wait and does not remove a keyholder. Asking to end protection has its own route, and it goes through a request to end protection.
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