In stars, helium can sometimes be fused into carbon and heavier elements (in their final stages of life). why didn't the same fusion processes produce carbon and heavier elements in the early universe?

Because the early universe was so hot that even tough the conditions to create heavy elements were there, they will break apart immediately after their formation because it was too hot for the heavy elements to remain stable. It was only later after the universe cold down that elements weren’t break apart and the process of forming heavy elements could begin.