When I was restoring my 1958 TR3A from 1987 to 1990, I knew that the heater core had been leaking. I couldn't solder the leak. A TR friend in Wiltshire, UK (Thanks Jon) gave me one that he said didn't leak. There were two reasons that it didn't leak. Firstly, there were no leaks in the core and 2nd, it was so clogged with rust and gunk that there was no flow. I filled it with water at low flow (no pressure) from my garden hose and then sloshed it up and down. Then rust, etc, started to come out one end. So I did the other end. Same thing. Then about about 100 fills, slosh and drain at each end, I got the water to gently flow from the inlet to the outlet. Then I continued to shosh it till the water was clear. Yours may be blocked like this one was.
It has been in my TR since 1990 (97,000 miles) and while my dear wife says she gets cold, I think that it's performing about like it did when the car was new.
If you think about all the stories of the rust etc, that blocks the drain tap on the block above the starter and all around cylinder #4, why wouldn't the heater core also be filled with crud since it is fed from the same area.