You need to modify your root class a little bit,
so that it will contain a Set<Wrapped> instead of a Set<UUID>.
@XmlRootElement(name = "root")
public class Root {
    @XmlElementWrapper(name = "wrapper")
    @XmlElement(name = "wrapped")
    public Set<Wrapped> myWrappeds = new LinkedHashSet<>();
}
And you need a separate class for the <wrapped> elements.
Surprisingly for me, you don't need an @XmlJavaAdapter for id here, because JAXB already has a built-in converter between java.util.UUID and String.
public class Wrapped {
    @XmlAttribute(name = "id")
    public UUID id;
}
I have checked the above with this XML input file
<root>
    <wrapper>
        <wrapped id="550e8400-e29b-11d4-a716-446655440000"/>
        <wrapped id="550e8400-e29b-11d4-a716-446655440001"/>
    </wrapper>
</root>
and this main method which reproduces the original XML:
public static void main(String[] args) throws Exception {
    JAXBContext context = JAXBContext.newInstance(Root.class);
    Unmarshaller unmarshaller = context.createUnmarshaller();
    File file = new File("root.xml");
    Root root = (Root) unmarshaller.unmarshal(file);
    Marshaller marshaller = context.createMarshaller();
    marshaller.setProperty(Marshaller.JAXB_FORMATTED_OUTPUT, true);
    marshaller.marshal(root, System.out);
}