What concerns within its empire caused the British to want to avoid European entanglements during the interwar period?
In the years leading up to the First World War, the very imperialistic system that spawned the British Empire had begun to crumble. During the war, and especially in the interwar period, the system all but collapsed. While the years of the European imperialistic race in Africa and Asia had come and gone, the animosity left in its wake remained. This being said, the British did maintain colonial holdings after the First World War, but they were less expansive than they had been.
After the end of World War I, politicians had the ability to look back at the war and consider what developments had contributed to its outbreak. One of the primary conclusions they reached was that the extensive colonization across the globe and the competition for resources it fostered, created a tension in the European community. Ultimately, the British concluded that if the First World War resulted in part from its (and other European nations') colonial actions, not engaging in such endeavors could prevent such a conflict from arising again.
During the 1920s and 1930s, the troubles in the British colonial holdings began to escalate, drawing much of its attention to those countries. This tended to leave the British in a weakened condition in relation to other European nations. With their difficulties, the British no longer enjoyed the access to resources they had before World War I. In the case of an issue with another European nation, the British would encounter trouble.
There are a number of instances in which the British struggled to maintain colonial authority throughout the interwar period, particularly in Egypt and India, but also in areas throughout the Near East.