Question: How were non-Jews saved in the time before Jesus came to die?
Answer: To answer the question, "how were non-Jews saved?", three different time periods need to be in view...
First - From Adam until Israel
During the first time period, everyone was a non-Jew. It was only from the time of Jacob (Israel) that God specifically defined His covenant to be with the nation that would descend from Israel. This was really an extension of the covenant that God had established with Jacob's grandfather Abraham -- now a covenant only through the line of Abraham, Isaac, and Jacob. (God is often referred to in Scriptures as "the God of Abraham, Isaac, and Jacob"). During this second time frame, history was clearly split between Jews and Gentiles (non-Jews). In fact, even the third time frame also has a clear delineation between Jews and Gentiles which continues to this day.
In the first time period, God required sacrifices, with people through obedience showing their faith in God. Specifically they believed that God would provide the perfect sacrifice -- the Messiah -- who would pay for sin once and for all. Even as the second time frame began, until the giving of the law (at the time of Moses, about 400 years later) the sacrificial system from the first time period continued. With the law, the sacrificial system got a lot more elaborate, yet still with a goal of pointing people to the Perfect sacrifice to come. But, in each of these time frames, the sacrifice could not save the person offering it. The issue throughout was not the sacrifice, but faith in the Person to whom the sacrifice pointed. [The sacrifice was merely to be an act of obedience and faith].
This gets to how people are saved during the third time period... no different than the first two. Salvation comes through faith in Jesus Christ (the Messiah). While those prior to Jesus looked forward to His sacrifice, we look back to its' fulfillment.
Even as now people can play church and profess to be a Christian, yet not truly be (Matthew 13:24-30), the same was true for the time prior to Jesus (Isaiah 29:13). There were many Jews who went through the motions of doing the sacrifices, who's hearts were far away from God. Again, the issue was a heart issue -- of faith! As the apostle Paul put it, "not all who descended from Israel are Israel." Paul points out that all who truly had/have faith in God are the true descendants of Abraham, the man of faith.
Every person that has come to faith in Jesus Christ (or ever will) throughout all ages, does so because they have heard the word of God. God has to reveal Himself to them.
So for all time, from Adam until the end, God's words that "Salvation is found in no one else, for there is no other name under heaven given to men by which we must be saved. (Acts 4:12)", remains true. Past, present, and future (until Jesus comes again), salvation is through faith in Jesus Christ [John 14:6, John 6:44, Ephesians 2:8-9]
And yes, there was a process by which a Gentile could join Israel and come under the law. During the time the law was in place a Gentile could 'convert' and come under the law, needing to be circumcised. This brought them into the external covenant of the physical nation, but as for truly being Israel (as Paul might have put it), they still needed to have faith in the coming Messiah. For these people of faith, the deeds were motivated by their faith and love for God. More detail on the ritual that was associated with conversion of a Gentile into Judaism is known from extrabiblical writings. Consider this excerpt from the Evangelical Dictionary of Biblical Theology (1996, Baker Books)...
The Bible's focus was primarily on the male convert and his male family being circumcised...
Read also the article on God's chosen people found here.