All this wifi detection nonsense is just a smokescreen. The real method is plain and simple server logs lol. These detection vans are just a decoy.
The BBC can certainly tell "someone" is watching iPlayer, it is tied to an individual device because of unique identifiers, and then this device can then be tied to an individual person.
They even know what devices you're using it on and how many different devices you watch iplayer on, they know how much you watch, they even know where you pause. It's also worth noting that identifying devices behind a NAT is one of the mandates of snooper's charter.
They log IP, what file you asked for, the time and browser/client type. If they could easily check server logs those letters would have the details in them to back them up.