Having registered a death, a registrar will, unless notified, subsequently make enquiries concerning the disposal of the body. Normally, the burial ground or crematorium will notify the registrar. See 5 and 6 here.

You will not be required to sign an undertaking to dispose of the body; there is just an assumption that you will.

If you fail to dispose of the body in good time you may face prosecution for preventing a lawful and decent burial (cf Rausing) and/or conspiracy to prevent burial. If the body is decomposing, they might have a point; as they would, also, if you were displaying the body in a manner likely to outrage public decency. But they wouldn’t if you are caring for the dead body in other ways. See Alternatives to conventional disposal question below.