Before assuming the worst, check you are using the right key. Locking sets are made in dozens of patterns and the keys look near enough identical, so in a household with two cars they get mixed up constantly, and a key from the other car will sit on the nut looking convincing and turn nothing. It is a five second thing to rule out and it accounts for a surprising number of the calls that come in from around Warwick Square and St George's Square.
If it genuinely is your key and the nut is not moving, there are two roads. You can find the code stamped on the key and order a replacement, which takes days and only helps if the pattern is merely worn rather than the nut seized. Or the nut comes out with an extraction socket in about ten minutes at the kerb and a plain one goes back on. Given that a car in SW1V is not going anywhere useful in the meantime, and the parking bay it is in is worth keeping, most people choose the second.
It is worth saying that this is not an unusual call. We come out with the extraction kits to Warwick Square, St George's Square or anywhere in between, get the nut off where the car is standing with the alloy protected, and then deal with the tyre itself. Ring at 3am from the Dolphin Square area and someone answers; Lupus Street at noon is no different. A 4am Belgrave Road call and a noon Warwick Square call reach the same fitter. SW1V takes in St George's Square and Churchill Gardens, and we work the lot.

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The grid between Lupus Street and Grosvenor Road is almost entirely controlled parking, so the real cost of a garage trip here is the bay you come back to find gone. A seized wheel nut is not worth that. The extraction gear comes to the car, the nut comes out without touching the alloy, and the wheel is left so ordinary tools will lift it.
A driver on Lupus Street was convinced four nuts had seized. It was the wrong key out of the wrong glovebox. The right one was in the house all along.
A car near Churchill Gardens had a replacement key on order while the tyre change waited. Extraction settled it in an afternoon and the key was never needed.
The car could not have reached a garage that week even if the wheel had come off. The nut was drawn where it stood and the bay was never given up.
It should push fully onto the nut with no wobble and no gap. If it sits proud, spins loosely or feels vague, it is either the wrong key or a worn one.
Only if the nut is sound and the key is simply worn out, and you can wait. If the nut has corroded, a perfect new key will not shift it either.
Around ten minutes for a single nut once the fitter is with you, and under an hour for a full set on most cars.
The car never moves, so yes. That is a large part of why people in SW1V ring rather than trying to get to a garage.
Not if you would rather not. A fresh locking set can go on instead, and either way it is agreed before any work starts.
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