I confirmed the server does have a .password file present for the pywikibot software login process to pick up on. And it contains entries for both CommonsDelinker and CommonsDelinkerHelper (both set to the same password).
However as both the public logs (https://toolserver.org/~delinker/helper.txt, https://toolserver.org/~delinker/delinker.txt) and a manual attempt by me confirm, the password for CommonsDelinkerHelper is correct but CommonsDelinker has a different password
Without this password or otherwise access to the account, I can't do anything.
I checked with Wikimedia to see what e-mail addresses these accounts are set to. Turns out neither are set to email@example.com. User:CommonsDelinkerHelper has its CentralAuth account set to Bryan's personal gmail address and User:CommonsDelinker is set to a generic gmail address for the CommonsDelinker project.
Per Toolserver's "multi-maintainer project" features, I've set up a .forward file in the home directory of the delinker account and added myself to it (Siebrand, Bryan, feel free to SSH into Toolserver, become delinker and add your @toolserver.org alias to the .forward file).
I've logged into Meta-Wiki as CommonsDelinkerHelper and changed its e-mailaddress to firstname.lastname@example.org and the confirmation mail has been delivered and dealt with.
What we need now is to figure out who and/or why the password of the CommonsDelinker account was changed and update the file and/or for that person to log in an change the address to email@example.com so we get the "Password reminder" mail and are able to change it to something we know and can configure the bot the server to use when ran from Toolserver.