The word, church, as with many words in the Bible, is used in various contexts and must be given distinct definitions. You mention the "church in the wilderness" as per Acts 7:38. I can also show you that the Old Testament saints were baptized as per 1 Corinthians 10:2 - "And were all baptized unto Moses in the cloud and in the sea." Therefore, we can see that baptism is not a New Testament creation either. However, neither the church nor baptism are found in the Old Testament by name. Therefore, the NT references to the OT events which use these words must have a special instructive quality. It is not necessary that they are equating the doctrine of the NT church and baptism with the OT nation of Israel or the crossing of the Red Sea. The church in the wilderness was a type of the NT church as the crossing of the Red Sea was a type of baptism.
When did the church begin? The Bible does not say. However, the word is exclusively a NT word. Also, it depends on what you mean by the church. The institution of the local church obviously existed during the ministry of Christ, else how could He give the advice to "tell it unto the church" (Matthew 18:17)? However, if you refer to the body of Christ (Ephesians 1:22-23) that includes all believers in heaven and in earth, this probably began after Christ left this world and sent the Comforter--the Holy Ghost--for it is the Spirit that places us into the body of Christ as per 1 Corinthians 12:13.