Didn't King David have women from other nations though?
I think you mean David's great grandfather, Yishay (Jesse), who married Ruth the moabite. But obviously, the process of conversion as we know it today is not as ancient as the Bible. I don't think that in the times of the Jewish kingdom, and certainly not in the preceding age of the judges, there was a formal process of conversion.
In later times, so called "tekufat hazugot" in the 1st century BCE, the conversion was formalized but it seems was still much easier process. This is evident for example by the story of the gentile who declared he will convert if and only if the entire Torah will be explained to him while he stands on just one foot. Shamai rejected him, but Hillel saw his sincerity and he told him:
"That which is hateful to you, do not do to your fellow. That is the whole Torah; the rest is the explanation; go and learn."