There is a key in the glovebox, so this ought to be simple. It goes over the nut, it turns, and nothing happens except a grinding noise. Locking wheel nut keys are cut to a pattern, and manufacturers use a wide range of them precisely so that one key does not open every car in the street. The one in your glovebox may belong to the previous car, or the previous set, or it may simply be a near miss.
A near miss is worse than no key at all. The splines on a locking nut are shallow, and a pattern that almost fits engages on the tips of them rather than the flanks. Lean on it and those tips fold flat, so a nut that was merely awkward becomes a nut with nothing left to grip. From there it is an extraction job whichever way it goes. If the key will not seat all the way home and turn with a solid feel, stop, because you are about to make it harder.
The key snapping in the socket is the one that surprises people. We come out to the car in SW16 with removal kits that bite on the nut body itself rather than the pattern, extract it, and protect the alloy while doing so. School run in Green Lane or 1am in Pollards Hill, we answer the same. Christmas Day in Pollards Hill, any day in Norbury: the line is open.

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Cars parked at the kerb all week in SW16 collect salt and grit around the nuts, and a locking nut that has not been moved since the last tyre change is the one that catches people out. There is no off street parking to work in along most of these streets, so the job happens where the car stands. Extraction, a check on the stud, then the tyre work carries on.
A driver had collected a second key somewhere along the way and neither would seat properly. The code stamped on the nut face matched neither of them. The nut came off with an extraction socket, the stud was checked, and a plain nut went on so the wheel could be removed.
Somebody had managed a quarter turn out of a nearly matching key before it slipped. That quarter turn flattened the splines and also nipped the nut a fraction tighter, which is the worst combination going. It still came off, just not the easy way.
A locking set usually arrives with a card carrying a code, and that code is what gets a replacement key cut. Cards live in handbooks and glove boxes and disappear along with previous owners. Photograph yours before the day you need it.
There is no such thing, and anything sold as one is a socket that bites the nut and destroys it, which is what we bring anyway. Patterns are not interchangeable between makes, or even between sets from the same maker.
That is common and it is not a disaster. The broken piece is drawn out or driven through, the nut is extracted, and the wheel comes off. It adds work rather than changing the outcome.
The tool grips the nut, not the wheel, and the face of the alloy stays protected while it turns. Recessed nuts down in deep wells are the fiddly ones, and they call for patience rather than force.
You can, if you still have the card and the time to wait for one. When the tyre is flat now and the car is parked on a Norbury street, extraction is the answer that has you moving tonight.
A plain nut with the correct seat, or a fresh locking set if you want the deterrent back. Either way it is torqued to the manufacturer's figure rather than nipped up and left to settle.
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