Thanks for this but I am using Apache. I did try restarting the server. Works OK for a few minutes then back to this annoying slow routine. Thoughts on that?

Maybe your issue is due to the connection being throttling above the server. However if this were the case the logs on the server wouldn't show anything to reflect the throttle so this is probably not the issue.

Have you edited Apache config files in any way?

Also check package install logs here: /var/log/apt to see if anything has been installed or updated since the issue started occurring.

Has this issue always been occurring or did it start recently? Maybe the server is actually overloaded. Run "uptime" immediately after getting the error and see what the server resources look like.

Also try to connect on another connection (for example a cellular connection) and see if the 503 error occurs on multiple connections at the same time or just from one connection in particular.

Assuming you are using Roundcube one could try upgrading Roundcube to the newest version.

You could additionally create a temporary test Apache virtual host pointing to the Roundcube location to see if can replicate on a new fresh virtual host on a temporary unique port. If cannot replicate the issue obviously lies in the Apache config files somewhere.

Would still try to replicate using the same principal of looking at SOGO on a new temporary virtual host on a unique port. At least can pin down the issue to Apache as a whole/ SOGO config files or the Apache config files that pertain only to the default virtual host.

Maybe someone will additionally have a better suggestion.

Seems no enough Apache child process?

What's the error message in both Apache/SOGo/Dovecot log files?