Merging has to be done by staff, following the procedure laid out in the Help (although it assumes that the later account was created accidentally).
Create and register an account, and then while logged in use the contact form, selecting "I need to merge user profiles". Fill in the form, giving as much information as possible.
The help panel says "We will reach out to confirm your ownership of each account, then complete your merge," which does indicate that SE staff will email both attached addresses to confirm. That might be a problem. But there is a box to explain the circumstances.
It would definitely help if the user can provide details which are not publicly available, for example the approximate date and reason for contacting staff in the past, or any moderator messages they received, or suspensions, or even [possibly] the content of deleted comments.
Because it's all done by staff, they will either decide that they have enough confidence that the accounts are owned by the same person, or that they don't. Since it would be catastrophic to merge two accounts where the requester is actually impersonating the other account-holder and claiming that no communications will be successful, I expect they will err on the side of caution: it's a high bar to get over. Providing details which only the account-holder can know can only help (although it's no guarantee of success).