Kingdom of Heaven was used as an idiom for Kingdom of God meaning God’s kingdom on earth.
The Jewish Kings and the Roman emperor didn’t take kindly to people who walked around Judea talking about setting up earthly kingdoms. Yet Jesus is a King and he did just that, set up a kingdom on earth, the Church. So “Kingdom of Heaven” was used instead of Kingdom of God. It was safer that way.
In fact, Heaven, the altar, the temple, the gold in the temple - these were all used in place of the word God, especially when swearing. One would swear on the temple or the altar or the gold in the temple instead of involking God’s name in a rash oath or as part of a curse.
When the people saw the sign he had done, they said, “This is truly the Prophet, the one who is to come into the world.” Since Jesus knew that they were going to come and carry him off to make him king, he withdrew again to the mountain alone. (John 6:14-15)
Jesus was careful about the perception of him as an earthly king and he doesn’t take sides. They tried to trap him by asking him if it was permissable to pay the tax to Ceasar or not. If he said no, he was against Rome and treasonous. If he said yes, he was against Israel and blasphemous. But he chooses no earthly kingdom, instead saying that one is to render to ceaser what is Ceasars and to God what is God’s. Jesus says, “I am not on the side of Isreal or on the side of Rome. I am on God’s side.”
Another parable he put before them, saying, “The kingdom of heaven is like a grain of mustard seed which a man took and sowed in his field; it is the smallest of all seeds, but when it has grown it is the greatest of shrubs and becomes a tree, so that the birds of the air come and make nests in its branches.” (Matthew 13:31-32)
Jesus describes the “Kingdom of Heaven” in his parables (Matthew 13) in different ways including the idea a mustard seed which turns into a shrub, and then grows into a tree. The reference to the tree is important and he even adds that birds will nest in the tree. Trees often represent powerful nations in the Bible, such as the “Cedars of Lebanon” and “a tree planted by running water” and the plane tree and palms listed in the Psalms and Song of Songs. “
The kingdom of Heaven is like a seed which grows into a tree” would have been an allusion to his own Kingdom growing into a powerful nation. Gentile nations were often represented by birds and these build their nest in the mustard tree - make their home in Jesus’ kindom which is the Church.
All of this is a way for him to speak about his kingdom on earth without actually speaking about his kingdom on earth.
-Tim-