Read the quote at the bottom of Beccy's post. That hole has gotten pretty deep.
Your "plan" is more like a prayer for a miracle. Miracles DO happen, but not that often to rescue us from our own actions.
Yes, it will be difficult (and embarrassing) to admit, but seriously, are you going to starve to death? Die from exposure to the elements? I don't live in the UK, but my understanding is that there is a pretty effective social services network there.
My point is, if this whole thing implodes at the end, you've only postponed the inevitable and perhaps harmed your work reputation far more than you would if you admitted the problem in time for the client to make other arrangements.