The first Minis, produced 1959-1967, were the Morris Mini-Minor and the Austin Seven, the latter a long-time Austin model name. Two years later, BMC launched Mini-based cars for two of its lesser brands. These were the Riley Elf and Wolseley Hornet.
Although there may have been some mechanical differences between these various Mini-based brands, from a styling perspective it seems to have been a matter of what is called "badge engineering" -- cheaply implemented superficial differences intended to preserve brand identities.
The original Mini concept was successful in its day, but in the long run BMC, Austin, Morris, Riley and Wolseley all disappeared from the automotive scene.
By far the most popular versions of British Minis were from Morris and Austin.
The Elf received a small bustle back and extended fenders. The hood and lower front fender cuts lines are carried over from Morris and Austin. A small version of a traditional Riley grille was added, along with "whisker" openings as part of the lower light ensemble.
Wolseley Hornets received treatments similar to the Elfs, the grille being characteristic of Wolseley. One difference seen on the car illustrated here is the smoothed front fender -- the angled sheet metal join is missing.