The question of which tiara was worn by Caroline Dewar in 1956 is a puzzle.
I wonder if the Duchess of Connaught's fringe tiara would have been at the disposal of Princess Arthur of Connaught and her son the future Duke of Fife in 1956.
There is no reason to suspect that the tiara did not also belong to Princess Patricia, especially as Princess Alexandra (Princess Arthur of Connaught)) had been given her own fringe tiara when she married in 1913. (And the Duchess of Connaught did not die until 1917).
If Princess Patricia owned the Connaught fringe then the bride, Caroline Dewar, would have been more likely to have worn the fringe belonging to Princess Arthur or the fringe belonging to Princess Louise, Duchess of Fife, which I note from information above was not auctioned until 1959.
When I enlarged Malluu's photo it seemed to me that the tiara looked more like the one belonging to Princess Louise, Duchess of Fife (rather than the one belonging to Princess Arthur), although it is impossible to be certain given the quality of the photo.
As Laura has already argued, the bride may have worn a tiara belonging to her family.