So for the player to change the base weapon "speed" you must Unequip then equip for the new speed to take effect. At least that is true for bows.
Annoying but the work around is simple.

However will this then effect all the NPC as well?

Say the NPC has a bow and the player has the same bow. Will changing the speed then equip swapping it on the player change the animation for all actors (NPC and Player?)

What if an NPC script changes the base weapon speed but the player is NOT using the bow and instead using say a sword.

Will the speed change work for the NPC without the equip swapping on the NPC?
If we equip swapping on that NPC then will it work for the player without equip swapping on the player if the player?

I know that is a lot of IFs...I suppose my main question that I really want to know is will changing the speed then equip swapping it on the player change the animation for all actors (NPC and Player?)

Anyone know for sure?

I don't personally know. Your best bet would probably be to set up a test-case mod that exercises many of the cases that you're trying to figure out.