Alright, so Jewel guided me through the issue and we solved the issue with ease – here is what was wrong:
Make the message dispatch
In the Contact menu in WordPress we regarded the ‘Mail’ tab
I had set the From email to ‘no-reply@’ but after changing it into ‘wordpress@’ it started sending.
Make the data go through:
In the Contact menu in WordPress we regarded the ‘Form’ and ‘Mail’ tabs
If your fields are set like this: [tel* Phone] in the Form tab, then you are required to use capital P as in the code – otherwise it will not be recognized.