If you look at the declaration you sign at the end of the form when you claim Benefits it says that you will report any change of circumstances promptly and that they have the power to enforce a civil penalty if not . So to be on the safe side I would report your change of address and partner as soon as you can. It says on the form you can report the change by phone or in writing.
Hope this helps, David
Nothing on this board constitutes legal advice - always consult a professional about specific problems