Cooling fan stopped working, changed the ect, now temp guage shows overheating in very few minutes. What next?

First the dash gauge is not that accurate. Yes if you have excessive back pressure from the catalytic converter then the temperature can rise on the engine.

The head gaskets blown

If the engine temp exceeds the limit the computer can kill the ac. If the cooling fan stops working the engine temp can go up and also stop the ac from being cold.
Do you know if the cooling fan is working when the problems happen ?

Is the car shutting down from overheating? Do you hear the cooling fans coming on at all?

One thing that a lot of people overlook is a sensor that tells the dashboard guage what the temperature of teh coolant is. I have had one go bad on me and my temp gauge told me I was way into the Hot zone within a minute or two of driving the car.

First question is what constitutes "overheating".
Since the car has been parked for years, the thermostat could be sticking closed.
The cooling system could have air pockets and need to be bled.
The water pump may be corroded inside and not pumping enough volume.
If the gauge goes above half way, the fans should start up. But half way should be more or less normal, not hot yet.
I would look at flow problems first, beginning with removing the thermostat and driving it without one for a few days. Also check and see if there is a process to bleed the cooling system of air.

Try this YouTube video: https://www.youtube.com/watch?v=WRmBdgAViuI Not sure, but hope this helps.

check your radiator fan if its powering on when engine runs hot. Test if there is power at the fan's electrical connector when car is running and temp guage is at half. If there isnt any power on, check your relays and fuses. Hopefully this helps

Had the exact problem. Turned out that it was the thermostat that needed replacing, relatively cheap and quick by mobile mechanic. Things were fine for a year and the second time it happened I assumed it was the same problem. However, this time the thermostat was fine, wasn't sticking, and I had a slow leak in radiator, losing colant and causing engine to overheat, which paradoxically had stopped the heater working as well. Had to get new hose also by local mobile mechanic and everything seems okay again except for the intermittent "stuck in park" problem that a lot of people are also experiencing.

Just fixed this problem. Replace the sensor or switch on the thermostat housing and viola!

The fans only run when the engine reaches approxminatly 235 degrees F. on the guage that is near the end of scale just before guage show "HOT" I don't think you have a problem, allot of folks out there don't know about this.