If someone has an approved DD 368 and doesn’t enlist into another branch. What happens to them then?

Just like the form says, once the DD 368 is approved, you have 180 days to provide a signed contract and new assignment orders to the approving authority so you can be discharged by the losing branch.
If you don't have them by day 170 or so, you can request an extension, otherwise the 368 expires on day 181 and nothing happens.

To further expand on what SGM Mccloud was saying, if it wasn't executed then you still are obligated to serve your existing contract. The conditional release does not negate the original contract