The convertible actually was a Daly Runabout built by Daly Cars of Edgware, greater London. Press releases were sent out in force in May 1982 to get the reports printed in time (the birthday took place on July 1). With these releases two black & white pictures were sent along, one of them showing the car with 'Broadfields' plate (the dealer?), while the other one showed only just the actual registration number, which may well have been 'TRF 34X'. That's at least what Daly Car's demonstrator and brochure car also wore. That car, however, was white and wore Weller wheels, while the Princess' car came with Mambas and may well have been a slightly darker hue? Silver grey perhaps?
Remarkably, no other image of Diana and the car are known and I wonder if she actually got it? One white Daly Runabout, which seemed exactly like the one seen in the brochure, was sold in 2016 and said to have been owned previously by cricketeer Duncan Fearnley. But while it also came on Mamba wheels, it wore registration 'MUY 68X'. DVLA doesn't recognize the number 'TRF 34X' anymore. Could it all have been one and the same car after all? Or were they two different cars in the end with the number plates swapped around..? But where is the car then, that was supposedly given to Lady Di..?
UPDATE 1: One reader with better eyes than me pointed out the difference in side stripes and their width. Closer inspection reveals that the striping on the Lady Di car is equally wide but in other and more contrasting colours, difficult to see on the b&w pictures. On the white car it's light blue and darker blue, on the Diana car it seems almost white (top) and black (bottom). See added detail picture below.
UPDATE 2: More to indicate that TRF 34X is another car as Lady Diana's car very possibly wore another registration number after all. DVLA records show that the following numbers may well have been issued to sister cars, all registered as Austin Morrises and with 998 cc engines:
TRF 35X - Registered October 1981, black, on sorn since 2012
TRF 36X - Registered May 1982, green, untaxed since 1991
TRF 37X - Registered May 1982, white, untaxed since 1990
TRF 38X - Registered May 1982, beige, untaxed since 1988
TRF 39X - Registered March 1982, red, on sorn since 2013
TRF 40X - Registered October 1981, red, untaxed since 1991