AK
You only have to show that the form was with the DWP by the required date, not that it is with the correct department, sending it by Next Day delivery on even Recorded delivery would be sufficient as both track receipt by the DWP.
If your form has been returned outside of the required date then I am afraid you will have to wait and see how the DWP deal with it, I am afraid that if they apply the rules strictly then they will close your claim, hopefully they will accept the letters you included as proof of your efforts to return the form on time.
Gordon
Nothing on this board constitutes legal advice - always consult a professional about specific problems