Take a little time replacing the antifreeze and water in the engine. Also re-check after a short run. Don't forget to replace the antifreeze if you have replaced the water pump.
Take off the air filter assembly for access to the two bleed valves.
Refill the water reservoir - I used this device made up from an empty drinks bottle and old reservoir cap as you will need a head of water above the reservoir to fill the heater matrix.
There are two small black plastic bleed caps - on one of the heater pipes near the bulkhead and on the thermostat itself. Adjust the heater to supply maximum heat to ensure the heater valve is open, but keep the fan very low or the engine will never warm up!
Fill the reservoir, start the engine and remove the bleed caps one at a time. Close the caps when there are no more air bubbles coming out. Keep filling the filler until the reservoir is up to the full mark and allow the engine to run up to temperature; recheck the reservoir level and that there are no leaks. It can take a while to bleed so be patient and check after the first (very short) run.
I lost some of the water/antifreeze when I replaced the thermostat and replaced it by soft water from the tap. When I checked the (original) antifreeze was green in colour; the top-up I got from the Peugeot dealer was red so I didn't use it. When I checked I still had about 40% green antifreeze so should be OK for where I live in the UK.
However I found this statement on this website
Toyota uses a red antifreeze in many of its products and should not be confused with the orange type long life antifreezes. It is essentially the green type of antifreeze that contains red dye to give it the red colour.
The Peugeot Stores operative confirmed this was correct and to top up with the red as supplied.
This is the schematic for the engine cooling fan.
Most failures appear to be the fan itself.
This is the external relay for the engine cooling fan. Mounted on top of the radiator and shown back to front to make reading the number easier
See the radiator fan circuit out (above). From the circuit you can see that the radiator fan fuse is MF1 which is hidden at the back of the engine bay fusebox.
I think the overheating symbol on the instrument panel comes from a separate thermal switch on the right end (looking from the front) of the cylinder head, so should give an unambiguous indication. Overheating could be due to a failed fan (or circuit) or a thermostat failing to open.
The fan circuit can be tested using PP2000 (which is how I know my fans and circuits are OK). The temperature sensor controlling the ECU gives a reading on a code reader at the OBD port so it can be checked as OK.
The temperature sensor mounted on the thermostat housing has failed twice on my 1.6 petrol. But that shouldn't cause overheating - just a confused ECU.
From other comments either the fan itself or the control relay on top are the more common faults.