The Extra Attack feature of the Bladesinger allows you to replace one of your weapon attacks with casting a cantrip. The Eldritch Knight's War Magic feature checks if you have "use[d] your action to cast a cantrip." Technically, you have used your action to cast a cantrip, or at least part of your action. Can an Eldritch Knight 7 / Bladesinger Wizard 6 make a weapon attack and cast a cantrip with Bladesinger's Extra Attack and then make a weapon attack as a bonus action with Eldritch Knight's War Magic?
For completeness, the features state the following:
Bladesinger's Extra Attack:
Starting at 6th level, you can attack twice, instead of once, whenever you take the Attack action on your turn. Moreover, you can cast one of your cantrips in place of one of those attacks.
Eldritch Knight's War Magic:
Beginning at 7th level, when you use your action to cast a cantrip, you can make one weapon attack as a bonus action.