Craft Commerce not sending E-Mails immediately
Thanks to @PhilipBoomy for his answer.  The problem were missing cron jobs.

Additionally I want to add one more solution from the craft support team:

If you don't have the queue set up to run automatically with a cron job or worker, you can also try adding this to your order confirmation template:

{# run queue #}
{% do craft.app.queue.handleResponse %}
{{ craft.app.response.content|raw }}


Which will run the queue over Ajax on that page.