The FMC is tracking the aircraft in NAV direct to your "direct to" tracking point. It fails, and you select HDG on the MCP, so you are still on track, tracking direct to the point. You call ATC and inform them you are no longer RNP capable due equipment failure.
If it was me:
ATC ask "Do you require vectors?", they get a cheery "No thanks."
ATC ask "Do you require routing via VORs?", they get a cheery "No thanks."
If they want to provide vectors to meet
their operational requirements, due to traffic separation with a non-RNP aircraft in their sector, then of course that's fine.
Knowing I am on track, a quick VOR check just confirms the expected time over the next point ("confirms", as the PLOG is filled out, of course), and navigation continues as outlined above.
I would actually enjoy the diversion from the boring old normal, and have a pleasant time with a minor failure which can be easily dealt with entirely "in cockpit".
If it was you:
Apparently feeling unable to navigate your aircraft, you require Vectors - so a PAN call would be appropriate, to guarantee the extra handling and workload from ATC.
If it was Jinkster:
A MAYDAY call is appropriate, as he apparently feels that an FMC failure with him as pilot will result in "the aircraft laying in flames" if he doesn't get outside help.
In terms of navigating with a map. Yes, I carry a Jeppesen plotter (specifically designed to be used in cockpit with Jeppesen charts) and a Jeppesen CR-2 circular slide rule. The method for using a ruler and pencil and whatnot to navigate about the place is beyond this discussion (i.e. - as a (presumably) licensed pilot, you shouldn't have to ask !)