I'm really out of my depth now. Here is the situation:
CiviCRM version 5.36.0 Wordpress Plugin Hosted on Linux Server
Administer->System Settings->Outbound Mail: enteredSMTP-Server, Testmail is sent and arrives at destination.
View Contact->Actions->Send an Email: Mail is sent and arrives at destination.
Mailings->New Mailing->Testgroup: shows 2 recipients.
Send test (single) 2nd try: Mail is sent and arrives at destination.
Send test (Testgroup - same as final recipients): Mail is sent and arrives at destination.
Next->Send immediately->Submit Mailing: Mailing is listed as scheduled, Email does not arrive at destination.
The Mailing report doesn't show any bounces.
I have tried different recipient addresses (two different test groups) on different mail services. I have tried with two different Mail providers for outbound mail. Everything works except the final mailing. I have no idea what else to test.