Well, I would think it would depend on a lot of things. Not least, what the monarchy decides to do.
At present, the queen is head of the commonwealth, and monarch of Australia, Canada, New Zealand, etc. The UK becoming a republic wouldn't affect those things. So technically, unless the queen stood down or the Channel Islands changed their rules, she'd still be in power there.
Usually, the queen is represented by a governor in the regions - I can't see that this arrangement would need to change if the monarchy was abolished in Britain.
So, the constitution of theChannel Islands wouldn't change and the legislation and agreements governing relations with the mainland probably wouldn't be affected.