How would you rule the Eldritch Knight's War Magic ability with a Bladesinging's Extra Attack? Could I make an attack, replace one attack with a cantrip, like Booming Blade, and use my bonus action to make an attack?

TLDR: Yes, but there's an important nuance involved when the fighter reaches level 11.

There's a little to unpack here, so let's start with the relevant abilities for this question. Bladesinging Wizards (Bladesinger) get Extra Attack at level 6, Eldritch Knight Fighters get Extra Attacks at level 5.

The Bladesinger gets the following version of Extra Attack:

Extra Attack (Bladesinging)
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.

The Eldritch Knight gets the following version of Extra Attack, along with another ability called War Magic:

Extra Attack (Fighter)
Beginning at 5th level, you can attack twice, instead of once, whenever you take the Attack action on your turn.

The number of attacks increases to three when you reach 11th level in this class and to four when you reach 20th level in this class.

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.

War Magic is worded poorly in this situation because "cast a cantrip" is not an action. To cast a cantrip, you would take the "Cast a spell" action and choose a cantrip. The Bladesinger feature allows you to "cast a cantrip" with your attack action. So rules-as-written (RAW), you satisfied the requirements for War Magic by taking the Attack action. Rules-as-intended (RAI), I'm not so sure but I don't think it's game breaking.

As long as the Fighter part of the multiclass isn't level 11, I'd allow the character to use War Magic and make an attack as a bonus action.

So now we're going to assume the multiclass is a 11 Fighter/6 Wizard and the player takes the Attack action. Can the player make 3 attacks, replace one of those attacks with, say Booming Blade, and use their bonus action to make an additional attack?

TLDR: No, rules-as-written specificially say you can not do that but doesn't quite say what you can do.

Let's take a look at the multiclassing rules for Extra Attack:

Extra Attack
If you gain the Extra Attack class feature from more than one class, the features don't add together. You can't make more than two attacks with this feature unless it says you do (as the fighter's version of Extra Attack does). Similarly, the warlock's eldritch invocation Thirsting Blade doesn't give you additional attacks if you also have Extra Attack.

Rules-as-written (RAW) makes it clear the two instances of Extra Attack don't get added together but it doesn't explain how it should work when there are multiple instances of Extra Attack -- only how it shouldn't work, that they don't stack.

For good measure, let's take a look at the rules for Unarmored Defense, as it provides some context for making a decision based on the designer's intentions:

Unarmored Defense
If you already have the Unarmored Defense feature, you can't gain it again from another class.

Rules-as-written (RAW), it's pretty clear you would get the first instance of Unarmored Defense. If you started as Barbarian and multiclassed into Monk, you'd receive the Barbarian's Unarmored Defense. Most Dungeon Masters (DMs) would probably allow a player to choose which one they want to use (this is rules-as-intended so don't be upset if you don't get a similar ruling) and they might be flexibile enough to allow the player to change which one they use during a short or long rest. Natural Armors work mostly the same way.

The reason I brought all of this up is that I'd interpret the multiclassing rules for Extra Attack with some intention based off the context of the Unarmored Defense rules. This means that when a player declares the Attack action and they have multiple instances of Extra Attack (or similar features), I'd make them choose which instance of Extra Attack they'd like to use with that action (Bladesinging or Fighter).

Do you want to use the Bladesinger's Extra Attack? Great, you can spend your action and bonus action to make a total of two attacks and cast a cantrip. Do you want to use the Fighter's Extra Attack? Cool, make your three attacks as your action and you can use your bonus action for something else (Sorry War Magic).