At max they could ask their local MP's to query CIC who would ask your consent first (in the form of signed consent form) to query CIC.
I have been through that and i don't think it effectively changes any thing. CIC proceeds the way they normally do with an application.
In my case, it was delayed because of strikes / wp rules being rewritten.
So the only feedback received was citing the above reasons for delay and that it is "in process". No more specific details. So i dont really think there is any advantage.