I tested it on the iron golem boss with the "boom attack"
because it has exactly 10 sec cooldown ,
i take the boss drop weapon "blunderbuss" for the test because it also has a 10 sec cd.
And the boss attack is always at the same % ready as the player attack no matter what, with or without subclass.
I also tested your scenario with the fire elemental:
my unarmed attack hits when the fire elemental attack is at exactly 66%
and i have the exactly same result with and without the subclass its always at 66%
weapon switch on and off, or reloading the game, or reloading different save slots didnt fix anything either.
Also the attackspeed time shown in the player stats page doesnt change after class change to barbarian but this could be ofc only a visual bug of the HUD i guess, which is not as important as the functionality