I agree with CreamCloud0's response: No, these don't stack.
War Magic requires that you are using your action to cast a cantrip - that's a Magic Action in D&D 2024 terms.
Whereas Bladesinger is replacing an extra attack of the Attack Action as a cantrip.
And from a game balance perspective my answer is also no, as you're looking to cast more cantrips in a turn. That's very dodgy to me as cantrip damage is already intended to scale (albeit more slowly) similarly to multiple attacks. Cantrip + bonus action single weapon attack is reasonably balanced I would think. Attacks + replacing one of these with a Cantrip + potentially bonus action two-weapon fighting attacks is starting to get overpowered. Multiple cantrips + attacks ... absolutely not!
Yeah, but Crawford does not call it a magic action. He seems to see the war magic as 'if you cast a cantrip with your action' to mean if any 'main action' results in a cantrip casting, then you get the bonus action attack. which is fair since the War Magic does not call it a 'magic action' (is this not a new category rule for the oneEdition, so not may be even backward compatible with wording found in the previous edition?) ie. War Magic itself is not a feature that is an action. it's a trigger, that triggers anytime a cantrip gets casts with any main action.
additionally, you are now the second person thinking in terms of 2 cantrips? huh? under the old rules there is no way to get two cantrips out of the synergy of War Magic and Bladesinger feature. so...you are building 'refutations' that don't exist. i'm not familiar with oneDnD changes, so i'm guessing you are somehow mixing that in?
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