Technical Water boiling

Hello everybody, I'v been ownin' a 1954 Chevy Bel-Air, over here in Italy for almost 40 years now.
Lately water/liquid started to boil. I checked the thermostat, but it was ok. I even tried without thermistat, but water kepr boling.
So, I was enough sure somethin' happened to the head gasket. I went to my local car repair shop: they checked the head..and the mechanical test said the head was ok. Anyway they rectified it and they're goin' to put a new gasket on, of course.
Now, do you think this boiling is caused by a failing water pump ? Or radiator ?...
Any idea ? I never had water boling problems at all. Thanks so much for your help. Alberto

 

By boiling do you mean it’s overflowing? What does the temperature gauge read?

could be as simple as a the radiator cap

when it’s hot feel different parts of the radiator, are there cold spots?

 

Yes, it's overflowing. Unluckily my temperature gaude doesn't work....so I don't know.
Now the radiator is apart...

 

I think I am goin' to redone the radiator core...what do you think about it ?