Was the gospel given to the whole world at the start of the New Testament?
No, it was not. At the beginning of the New Testament era, the gospel was confined to Israel. Acts 11:19 records that those scattered by persecution preached the word to none but Jews only. The transition to Gentiles came later.
Les explains that the Book of Acts is a transitional book. He notes that after Peter's experience at Cornelius, Peter still did not go to the Gentiles; he understood he was to confine his ministry to the Jew, to the circumcision, and they agreed Paul was to go to the Gentiles (Galatians 2). Les points out that Acts 11:19 says the scattered believers preached the word to none but Jews only. He says this verse opened his eyes, because he had always thought that as soon as you got into the New Testament you were in Christianity and the whole world was being given the gospel. But he says, 'No it wasn't.' It had been confined to Israel, and even these Jewish believers had no concept of going to Gentiles. Les says chapter 11 shows the transition to the Gentile kicking into a higher gear, with the Jewish program slipping away.