There's a real world example of this situation from the six-month Irish bank strike of 1970. Cheques, pubs, and trust ultimately replaced the banking system; and everything kept moving; not perfectly, of course, but neither are our current systems.
A couple of links to get you going, but there's a lot of data out there on this:
https://archive.is/20180106184835/https://www.ft.com/content...
https://archive.ph/20120802230107/http://www.independent.ie/...
It's also an object lesson on how money is created out of thin air.