I got this Wndsor Carrera Sport (around 1973 from what I can find on it). I replaced the cables and adjusted the derailleurs. It has friction bar end shifters so the adjustment should have been no problem. I discovered the front has no problem changing from the small to the large when I’m on the flat or going down hills. However, when climbing, I cannot change from the small to the large. I have never tried to shift from small to large front gears, going up a hill, because there is generally no need to. I usually shift from the large to small just before the hill. Everything else works fine. My question is: is my adjustment correct or should I be able to shift small to large when going up a hill?

I lost the nut to the front brake lever and as soon as a replacement arrives in the mail, I’ll take it out and start riding it.

When the chain is under heavy tension, such as climbing, the front derailleur has a tough time moving it, in either direction.
Most FDs use the spring to shift from large ring to small, so you are at the mercy of spring strength. Shifting up, is based on your force on the shifter, and flex in the derailleur. Either way, ease up on the pedals while shifting the front.

It's not designed to shift from small to large chainring going up hill. As you said, there's no need to.