Yes you are allowed to wear it and its not a problem. As a prove you can test it out during the week by leaving it laying around for 24 or 25 hours without winding it and see if it would still work. This would prove that they are capable of working through out the whole shabbos without you winding it and so you aren’t doing anything by wearing them.
I’m sure there are some poskim who will disagree so you have to ask your LOR.