Posting this "issue", so that others may find the solution faster than me:
Requesting an account will trigger an "504 Gateway Time-out" error.
/var/log/nginx/error.log:
[..]upstream timed out (110: Connection timed out) while reading response header from upstream, client: XXXXX, server: XXXXX, request: "POST /w/Spezial:Benutzerkonto_beantragen HTTP/2.0", upstream: "fastcgi://unix:/run/php/php8.0-fpm.sock", host: "XXXXX", referrer: "XXXXX/w/Spezial:Benutzerkonto_beantragen"[..]
Solution: $wgSMTP was not working because of mailserver changes. Fixed SMTP settings and everything was working again.