You can respec anyone - including origin characters. Only two subclasses are locked for origin characters (Shadowheart?). Even if you respec Halsin he will still be a Druid. Same for the other Origins. It won't remove the tags.
Wyll and Shadowheart for sure. First, they are intimately tied to their sub-class; and second, they are the only ones that pick a sub-class at level one anyway.
It sounds like companion main classes are locked and two main subclasses are locked so maybe some some subclass are changeable...which as you say definitely Wyll and Shadowheart. Though I'm not sure if it would make sense to change Minthara's Paladin subclass...which I'm not sure what subclass she is but surely there's no way we can put her under a proper oath.
