Can you clarify which Nintex Workflow platform you are using?

I am using Nintex on Microsoft SharePoint 2013. Sry, that I didn't mention that before.

how do you assign the task? to the specific user or to a group?

if assigned to a group, all group members should be able to respond.

if assigned to specific user, you can not grant permissions to respond, you have to delegate the task to other person/group.

Hi!

I assigned the task to one specific user who is responsible for the task. But if this user is not available or only available per phone it should be possible to "manually" approve the task be the owners of the process.

The delegation is again only possible by the user to whome the task is assigned to, right?

there are several possibilities how to escalate task.

if you want to do it manually, site admin can delegate any existing task.

there are as well automated ways how to delegate task, eg:

- user can set up in his profile to delegate all the task if he is out of office

- you can configure an escalation in task action in workflow that delegates the task

- there is dedicated 'delegate task' action

- you make web service call to delegate a task

not sure which way might suit you the most.

see also Cassy Freeman‌'s summary on all the possible delegations  

I am Site Owner and cannot find the option to delegate the task.

Furthermore I don't understand why i can approve tasks that are created within a SharePoint 2013 workflow (with SharePoint Designer) but not those who are created within Nintex.

you have to delegate the task from task form.

so navigate to the task form either directly from workflow task list or through item/workflow.

Sry, for asking again, but I cannot find the option to delegate the Task in the Task form of the Task.

"Allow Delegation" is ticked in the workflow Task where the task is created. But if I open the Task from the task list, I cannot find the delegation option. It's neither in the menu, nor somewhere at the form.

by default, there is placed a text control on task form that provides a link to delegate task.

if you have customized task form and removed the link/control you will have to re-add it there on your own.

delegate URL link is provided by 'Delegate URL' common reference.