You might want to start from a fresh point of view. The OP's question is flawed to begin with. (Not blaming the OP for that - nor is even the WT the originator of the flaw.)
Rev 16:16 describes Armageddon as a place (Gk topos - from which we get "topography"). You usually do not speak of, 'when does some place begin.' Although, sometimes events (including battles) are referred to by the name of the place they occurred (Waterloo, Normandy, et al).
The context of Rev 16 has 'kings and their armies' gathered to a place (vs. 16), then God saying 'it is finished' (vs. 17), then a series of events occur (vss. 18-21). (All/each of the events are preceded by "and" (Gk kai - a conjunction), in a pattern that looks much like a rhetorical device called polysyndeton. Each conjunction in the list separates each item.
As an example of one of the WT's flaws in interpretation, they take the first phrase of Rev 16:19 ("And the great city was split/became into three") and say it is describing the third phrase of that same verse ('And Babylon the Great was handed the cup of God's wrath'). Under the rhetorical device of polysyndeton, each of these phrases would be an event separate from the other.
In other words, Armageddon is a "place" where several related (as a result of God's saying 'it is finished') events occur. Using the following descriptive explanations in Rev chapters 17-19, the "war" that occurs, as described in Rev 19:11-21, equates with the metaphors described in Rev 16:20, 21. (Where the removal of mountains and islands (Rev 16:20) would picture the defeat of the kings and their armies in Rev 19:19, 20, and the people being pelted with hailstones (Rev 16:21) would equate with 'the rest being killed off' in Rev 19:21.
The 'kings and their armies being gathered to Armageddon' would occur before God has said "it is finished."