Q: When was the prayer shawl worn by Jesus and when is it worn today?
A: There is no record of Jesus ever wearing a prayer shawl and futhermore the prayer shawl did not become a major item until beyond the first century Israel. The wearing of the prayer shawl was a rabbinic innovation which came after that time but did not exist at the time of Jesus. What Jesus did wear were the tassels on the corners of the garments in keeping with the Mosaic Law but these were tied to the garments themselves and were not a separate prayer shawl that one put on for the morning prayers. Jesus Himself never wore a prayer shawl nor was a prayer shawl ever mentioned.
As to when it is worn, in Orthodox Judaism it is only worn in the morning prayer services whether it is at home or in the synagogue, but never worn in the evening except on two occasions: the Day of Atonement and the Feast of the Ninth of Av, commemorating the date when the first and second Temple were destroyed.