Military First will boost your military units' training speed significantly. Like all other policy cards it obsoletes - Agoge, Feudal Contract and Grande Armée - it can be very useful for those aiming for a Domination Victory to start prepping up for a (very) late-game war to overthrow the last few civilizations that remain after the early-game conquests.
However, civilizations reacting to a war declared by a stronger enemy could also use the Policy effectively, and come up with armies of Mechanized Infantry where before there were only a few Machine Guns for defense.
Note that the Production bonus does not apply to siege units! So if you are the defending side and suddenly you are now counter-attacking, this policy will not be useful for training machines to take down enemy walls. It doesn't apply to light and heavy cavalry either, so if you would now like to pillage everything in retribution, Lightning Warfare is suggested.
When the United States entered World War II, it was woefully unprepared, and it took months to put America on a war footing. As a result of the Cold War, the administration adopted an amorphous strategy of “Military First” for spending and research, leading to an informal alliance between military and corporate interests in influencing public policy. In his farewell address in January 1961, President Eisenhower first referred to the “military-industrial complex.” Currently, U.S. military expenditures account for about 47% of the world’s total arms production, and in 2011 the United States spent more on its military than the next 13 nations combined. “Military First” indeed.