You updated your question to talk about the time before Matan Torah and the time after Matan Torah. Technically, the entire nation "converted" by accepting the torah and became a "religion" as well as a nation. After that, if a person wanted to join Bnai Yisrael, they had to go through the formal procedure before a bais din, go to the mikavah, and if a male, have a bris milah. Before then thre was no official bais din, but we see that they had to join the family of Avraham Avinu, which implies accepting Hashem and having bris milah, but in the way that Avraham converted the men and Sarah converted the women.
That is, before matan torah, just as the avos accepted the mitzvos as "ainah metzuva ve-osa", so too their students and those who married into the family accepted it on that basis. After matan torah, they had to undergo conversion in the same way that Bnai Yisrael did at Har Sinai. We learn the methodology from Rus. Technically, from that time on conversion was the same way as we do it today.
We learn this, not only from Rus, but from the stories of Hillel and Shamai and the geirim that Shamai rejected because he thought they were mocking Judaism and Hillel accepted because he saw how to bring them in.