I believe it DOES apply extra attacks under the following understanding:
"...when you take your (regular) action to cast a cantrip, you may make one weapon attack as a bonus action."
You're allowed to make an attack action as a bonus action, it just must be made with a weapon. So in this case, you cannot make an unarmed attack with this bonus action. Although you can make the attack with a bow if you so chose, since it doesn't specify past "weapon attack".
"...you can attack twice, instead of once, whenever you take the Attack action on your turn."
Multi Attack says "instead of". So rather than being limited to a single attack (as the attack action is described in PHB), it would allow as many attacks as your fighter level allows. A standard attack action is defined as "...whether you are swinging a sword, firing an arrow from your bow or brawling with your fists" (according to PHB). Two of these are explicitly making weapon attacks and War Magic specifies "weapon attack". Swords and Bows are both weapons.
I also want to cite several spells that do not count as regular actions yet would undoubtedly count as "casting a spell":
- Shield (1 reaction)
- Magic Weapon (1 bonus action)
- Counter Spell (1 reaction)
Even though these are clearly spells not using the regular action, they are spells and all are subject to consuming spell slots when cast. On this same logic, War Magic is an attack action taken with the bonus action and simply requires that said attack action is made with a weapon.
I know that most of the answers on this topic so far say that this isn't taking an attack action, but my logic comes from the fact that some spells specify when they take an action other than the usual normal action yet still count as "cast a spell". Despite the attack requiring a weapon, it should still count as "taking the attack action" even though it's using the bonus action and has a prerequisite to using a cantrip beforehand as the regular action.