Hey there SmoothCat! Welcome to PF!
If it flies off at the same speed as its tangential speed just before
… so its angular momentum (about the centre) stays the same … then no, there's no change in angular velocity of the rest.
That's because if the angular momentum of the detached part stays the same, then so must the angular momentum of the rest.
(I'm sorry, but any more detailed response would be superfluous!