When will the kingdoms of this world become the kingdoms of the Lord and of his Christ?
They become the Lord's and His Christ's at Christ's return, when He sets up His kingdom on this planet. This occurs as His thousand-year reign is established, an event described as fast approaching, and He shall reign forever and ever.
Les explains that Satan offered Christ the kingdoms of this world if He would fall down and worship him, but Satan evidently did not understand that Christ was going to have those kingdoms. They are going to be His. Daniel says He will obliterate them from view, and they will blow away like the dust of the summer threshing floor. Christ will set up His own glorious kingdom; His power will be absolute, total, and glorious. Since Christ's return to set up His kingdom is fast approaching, Revelation 11 says the kingdoms of this world are become the kingdoms of our Lord and of His Christ, and He shall reign forever and ever. Les feels the thousand-year reign delineated in Revelation will be the introduction to the eternal; after the thousand years run their course on the planet as we now know it, it too will be destroyed.
And the seventh angel sounded; and there were great voices in heaven, saying, The kingdoms of this world are become the kingdoms of our Lord, and of his Christ; and he shall reign for ever and ever.