I would like tp help, but your question is oddly phrased. Historiographic refers to the methods of analysis and writing of history; most often discussed in terms of theories of analysis and what conclusions those different theories reach. So are really you asking how authors over time have analyzed the British in WWII? Or perhaps just interested in the general consenus about British shortcomings in the war.

Winston Churchill's diaries about the war is a good place to start, and so are writings of military historian extraordaire John Keegan. Both mention the lack of preparedness, poor leadership, and poor interaction with dominions and Allies early in the war. However, when Churchill became Prime Minister, things steadily improved. The appointment of Montgomery as C.O. in Africa had a great impact for the better. "Monty" was great a the set piece battle and waited for overwhelming force to make a move. He stressed cooperation and coordination of all combat arms. Whether this tactical approach was good or bad depends on the situation. Monty had the benefit of airpower, large reserves and steady supply. In Italy, his leadership suffered, because he could not get cooperation from the other Allies, and the Germans executed some their best defensive fighting in the war. Placed in charge of D-day planning and operations, having the support of Eisenhower to ensure cooperation, he made sure the Allies coordinated their efforts. As the Allies broke out of Normandy and pushed the Germans back, he played the risky airborne operation Market Garden, which went against many of the elements that brought him success: solid intelligence, superior force, and effective coordination. Monty and the Brits learned from the mistakes and made solid gains across the Rhine after the Germans depleted their strength in the Battle of the Bulge.