The difinitive part of your post is that the heater isn't hot. A stuck shut thermostat would still provide heat to the heater. A water pump that has shelled the plastic impeller will not pump the water into the heater so the heater is cold.
Something to remember though is that overheating can damage an otherwise good thermostat. Contiques generally don't need a thermostat when the water pump fails, but it would not hurt to change it because overheating often damages thermostats and you are nearly there anyway.
