Sorry for the few infos.
The designer mode is turned off.
There is another moodle instance on the same server with much more data that runs flawless.
Both are running on moodle 3.1 with the essential theme.
During the installation there were no problems nor errors. Further I haven't activated automated backups or statistics.
Sadly I can't remember when it started to take that much time to log in... I can only assume that it went longer during the customisation of the instance since I didn't noticed it first.
If any more information is needed I would be happy to provide it.