Thanks for your reply Marley_Ring.
I have checked the ring app and everythign is setup fine. This is confirmed for me also because when the ring video unit is restarted, everything works again. If there were problems with the setup in the app or on the individual chimes, this would not happen.
I will happily try unlinking the chimes and re-linking them but cannot see the logic of this as they work when the ring video unit is rebooted. For me, it sounds like a faulty ring video unit, which is a shame as it is just over a year old and out of warranty.