The only email header field that can be replaced is the Reply-to field? Under the heading E-mail header details in the e-mail settings for my webform I have set the option E-mail from address to the component E-mail, from the current webform.

When the webform is submitted and recieved it works to click Answer in the recipient's Outlook client (the user e-mail from the webform is used in the TO field of the new mail).

But unfortunately this does not work when the recipient have enabled an auto reply, such as Out of office. In that case the address used in the auto-reply mail is the site's default e-mail address. 

I have cleared all fields where the default FROM e-mail address can be stated (Default e-mail values in webform config, E-mail options in the SMTP settings) and tried with the option Use Reply-To header in the webform config both enabled and disabled.

I have also tried to use the variable $from instead of site_mail in htmlmail.tpl.php.

Is there really nothing I can do to replace the FROM address with a custom address fetched from the webform? Apparently the only header field that the webform can replace is the Reply-to field?
Not long after posting this question I found a working answer here
The problem can be solved programmatically using hook_mail_alter,