1961 Other Triumph Models Low oil Pressure

Engine Mechanical problem
1961 Other Triumph Models 4 cyl Two Wheel Drive Manual

My car is a TR3A and its year is actually 1958. It has many modern updates including an oil cooler. At start up oil pressure is 90/100 psi. But as the oil gets warmer the pressure soon drops and within 10 minutes or less, the pressure drops to 5/10 which, of course, operates the warning light which shines brightly at pressure below 20psi!
After a run the oil cooler itself is cold to the touch. The two pipes up to the thermostat are very hot and the two pipes from the thermostat to the oil cooler are, one hot and one only warm. Does that not indicate that the oil cooler is working?
What can be wrong? I have recently replaced the oil pump but it has made no difference.