I love puzzles.
So my first thought is that various programs get launched when a user logs on. I wonder if one of them is hanging when the second user logs on because the program is already running from the 1st user.
https://learn.microsoft.com/en-us/sysinternals/downloads/autoruns
Run autoruns and click on the Logon tab. You can uncheck entries which will temporarily disable that entry.
Start with the entries in HKLM\SOFTWARE\Microsoft\Windows\CurrentVersion\Run and HKCU\SOFTWARE\Microsoft\Windows\CurrentVersion\Run.
It won't solve the root problem but if disabling a specific entry makes the logon speed up, then you'll know where to look next.
If that doesn't find anything then next I would try Process Monitor.
https://learn.microsoft.com/en-us/sysinternals/downloads/procmon
That will show you every file, registry, and network call that your system makes. It can be a lot of data. I usually configure it to capture 1 or 2 million entries and to drop filtered events.
It's a tool that you have to play with a bit to figure out. You'll want to try to see what calls are being made during the 5 minute hang period. And what process is making them. You should be able to see when processes get launched by the second user.
After the second user gets logged on, immediately switch back to 1st one and stop the trace.