This story about the royal connection of this DS420 is doubtful. The Queen Mother never ordered DS420 cars in green colour, but always in so-called "Royal Claret" with or without black roof or bonnet. The "Queen Mother's Cars" page on this website clearly shows this.

There was no such colour as "her special green". The factory records of this car specify the body colour as "Moorland Green". Only one other DS420 was delivered in this colour, and that was 11 month before this car was produced.
That other "Moorland Green" car was sold via the dealer Thomas Startin in Birmingham (similar to this one). But cars ordered by the Royal Household were either bought via the Government purchase channels, or marked as "Special Delivery", or they simply stayed the property of Jaguar Cars Inc. and were thus not sold at all.

However, the Royal Household indeed had special arrangements with various owners of privately owned DS420 cars, in several large cities, to make such car available for Royal visitors. This car may very well have been one of those.