I thought it had used to stand for 'Associated Dairies' or was that another company?
From ASDA website:-
"The years leading up to the formation of Asda were years of momentous meetings, partnerships forged in the heat of inspiration, big ideas, great dreams and bold actions.
Its roots can be traced back to the 1920s and just two branches, but no-one knew then quite how incredible the future would be.
In the beginning
The Asquith family (W.R. Asquith) ran a butcher’s shop in Knottingly, West Yorkshire, which was eventually expanded to seven shops. The two sons of W.R. Asquith, Peter and Fred, were actively involved in the family business and were later to become co-founders of Asda.
At the same time during the 1920s, a group of enterprising West Riding dairy farmers joined forces under the banner of Hindell’s Dairy Farmers Ltd. These included the Stockdale family (A. Stockdale).
Through a process of acquisition and diversification, a new public company was formed in 1949 – Associated Dairies & Farm Stores Ltd, with Arthur Stockdale as Managing Director.
During the 1950s and early 1960s, Associated Dairies had expanded the number of pork butchery shops (under the Farm Stores fascia) and had also created the Craven Dairies brand for its cake shops and cafes. The son of Arthur Stockdale, Noel (later to become Sir Noel Stockdale) met and struck up an immediate rapport with the Asquith brothers. He went on to become the other co-founder of the future Asda, in a meeting that would change the way we shop forever."