At least 19 people, including five policemen, were killed in the blasts, which happened within minutes of each other in the central Shia district of Karrada.
Canon White, leader of St George’s Church in Baghdad, said that the children were on their way from the church after practising for their first Holy Communion service.
The service is one of the biggest occasions for the church and takes place each year at the end of August.
In the run-up to the service, the children attend church for around two hours of preparation each day. They are transported to and from St George’s by the church’s own bus.
Canon White explained that the children had been on their way home when they were caught up in the blasts.
"We are used to bad days, but today it has been in a different league,” he wrote on his Facebook page.
“As they were on their way home today they were caught up in a major bomb. The bus was not too badly damaged, and it continued it journey.
“All of children started singing the Lord's Prayer. As they were in the midst of their prayer, a huge bomb exploded. All the glass of the windows shattered, people fell to the floor injured, and one of the teachers was in a very bad way.
“They were all taken to hospital, but were all discharged as none of them had life threatening illnesses.”
Canon White was able to visit the children in the hospital.
He reported: "It was wonderful seeing all the children. Despite their injuries they were all smiling, their parents were not - some were in floods of tears on learning what happened.
"The fact is that today was miraculous. When you see the state of the bus it is simply a major miracle that nobody died or was more seriously injured.
"The children knew quite clearly why, they just kept telling me that they were alright because they were praying and G-d was with them, and He was.
"Many tears were shed, and there are huge things still to do and there are people who still need treatment.
"Please pray for us as we try and recover from another trauma."