Quick answer for engine overheating/cold radiator would be:
a) Check there is coolant in the system: refill as needed and check for leaks. Remember to purge air bubbles: my way is that as I add coolant I squeeze the hoses attached to the engine, like a hand rubber pump, then start the engine without radiator cap, for some seconds, then redo.
b) Check thermostat: an easy way is to remove the thermostat completely and run the engine without it for a test. A way to check the thermostat is to heat water and immerse the thermostat hanged with some wires, if water is hot enough (varies for thermostats but would be between 80C to 90C) the thermostat should "click", i.e., open. NOTE: some engine designs requires the thermostat to be placed in its correct position, otherwise coolant won't flow as designed/needed.
c) Check water pump: be sure engine has coolant, start engine and slowly release the "out" hose from the engine, commonly at the engine's head. Or slowly release the hose to the radiator. You should see water coming out with some pressure. Otherwise, remove water pump and check if it is sound.
d) Check water coming out from engine: the engine block/engine head could be severely stuck. Add coolant, start engine, release "out" hose. Coolant should flow out with pressure.
e) Check "in" and "out" hoses in radiator: release one first, check, then the other. Coolant should flow freely, otherwise radiator could be clogged.
Resuming what to blame:
No coolant, bad thermostat, bad water pump, clogged engine block/engine head (it was my latest ordeal...totally blocked with soil), clogged radiator (was my previous ordeal, when acquired this car...tons of gasketmaker silicon gunk).
When starting the engine for the procedures I said above, always do with cold engine! Don't scald yourself. Starting the engine for a few seconds, cold, to check this stuff won't be enough time to heat it that much, and if it does, just let is cool down.
The correct way to install the thermostat into its flange: