This is more of a "How would you DM this situation?" rather than a rules question. If someone were to use the Booming Blade or Green Flame Blade cantrip in an anti-magic area or against a creature with magic immunity, obviously the spell would fail. Would you tell the player, "You took the casting action and it didn't work. That's it for your action." or would you still allow the melee attack since it is part of the spell? Obviously, it would just be a normal melee attack, but would you allow it?

The melee attack is part of the action used to cast the spell, and a prerequisite for the spell's effect. In my opinion if they have an action available to cast the spell, they're entitled to the melee attack regardless of whether the spell effect goes off or not.

I agree. They (attempt to) cast the spell, but the spell has no effect or fails to cast. But the weapon attack was part of casting the spell, not part of the spell's effect so it still happens.