Times of the Gentiles The context of this verse deals with the Destruction of Judea in 67-70 A.D. "Trampling" deals with military warfare, not just foreign occupation of a country. Verse 20 sets the stage.
The Roman General, Cestius, came and surrounded Jerusalem with his armies, then mysteriously retreated. (Some say it was due to drought, and lack of water for his men.) This retreat gave the Christians time to flee to the mountains (21:21) Nero heard of the retreat, and the rebellion of the Zealots, and summarily dispatched general Vespasian to conquer Judea. (F. Josephus, Wars of the Jews)
Synoptic Gospel What Luke describes is the same as the Olivet Discourse in Matthew 24-25. What most people reading this passage don't realize is that Luke is not just talking about the siege of Jerusalem and the Destruction of the Temple. But the Roman trampling was the statement describing a 3 and 1/2 year conquest of all of Judea, from Dan to Beersheba! Vespasian burned down villages in Galilee. Samaria, Perea, Negeb, as well as Judea proper, year after year. THIS BEGAN IN THE SPRING OF 67 A.D. This was the beginning of the THE TIMES OF THE GENTILES, and it wasn't completed ("fulfilled") until the Fall of 70 A.D.
Gentile Army Notice the accuracy of the prophecy in Luke! The Roman army of Vespasian (and later Titus) consisted of 4 Legions from Egypt, Syria, Rome, etc. They were comprised, not only of Roman soldiers, but composed of several mercenary legionaires--many nations! So TIMES OF THE GENTILES (NATIONS) was right on.
THE TIMES The "Time" referred to was the Time in fulfillment of all that has been written (v. 22) Daniel had prophesied about this destruction in chapter twelve, giving the exact number of days this 3 and 1/2 year trampling would happen: 1290, extended to 1335 days. (Daniel 12:11-12) Amazing!
The "trampling" which began in the Spring of 67 and came to a close in the fall of 70 A.D. was hindered from completely destroying all of Jerusalem, by a hold-out of some stubborn Zealots for 45 days: 1290--->1335 days. Just like Daniel was told.
Subsequent History The history of the Jewish people and Judea has been one of tenuous existence, chased from country to country. They have been oppressed, and suppressed. But the "Trampling" of this Luke passage deals with the army invasion by Rome in the first century. Trampling is what armies do, and Rome certainly did it. Notice the admonitions to flee, etc. These applied to that first century Christian congregation (and listening Jews) who "were in Judea." (21:21)
For in depth detail see Flavius Josephus, "Wars of the Jews"; Raymond Grant, "Times, They Are A'Changing"; Marcellus Kik,"Matthew Twenty-four".