My current hypotheses about the War's impact (asserted but not here defended):
The War's influence on British philosophy was primarily negative -- it turned philosophers away from idealism. The negative impact was effected primarily in the generation of philosophers who came to maturity in the 1920s and '30s, and not so much in those who already held academic positions when the War began or who fought in the War.
One part of the War's negative impact was to challenge idealism's optimism about history. In this case, the War posed a challenge specifically to one of idealism's tenets.
For the most part, though, the War's negative impact was rooted a broader, cultural factor, namely, post-War Britain's antipathy towards Germany. It might seem insulting to claim that philosophers were strongly influenced in their philosophical thinking by such extra-philosophical factors; and, certainly, it would be insulting to claim that British philosophy changed direction because philosophers thought, 'Idealism? But that's positively Teutonic! Ick!'
The negative influence of anti-German sentiment in British philosophical circles was more subtle. It was more pronounced in those British philosophers who were too young to have known much (if anything) about idealism before the War. Thus, the negative impact was achieved not by prompting anyone to abandon a philosophical position but, instead, by means of (perhaps unconscious) biases that conditioned post-War British students' study of philosophy.
These biases affected students' tendencies to be more charitable to some authors than to others, to devote more time and care to the interpretation of some texts than to others, and so on. Such tendencies rely on something like good will towards (or trust in) the author, and these attitudes were in short supply when it came to German authors after the War.
Among the relevant biases was a patriotic proclivity for views that appeared to be more in line with British philosophical traditions (e.g., empiricism). Also, the animosity to all things German was, if anything, even more pronounced in relation to German academics, since so many thousands of German professors (inc. many philosophers) had signed proclamations defending Germany's cause and strategies in the early stages of the War. These proclamations made it easier for post-War British academics to regard their German counterparts as bombastic and confused (rather than simulating and profound). Indeed, the whole German tradition was impugned, and it became more plausible to suppose that the tangled thickets of Hegel and others (even Kant) were a mere tangle that was not worth the exploratory effort.
German idealism is more difficult to understand and appreciate than many other modern philosophies, and learning from it requires more patience and trust. After the War, British students no longer had this trust. As a result, there were prone to see little more than pretentiousness and obfuscation in the central texts of German idealism. The texts of British idealism (esp. those of Bradley) suffered a similar fate partly due to their association with the German idealists.
Update (Feb. 23): The anti-German bias was focused on philosophers who held positions in the German universities. They were more likely to have signed proclamations in defense of the German war effort, and their positions linked them more closely to the belligerent, enemy state. Central European outsiders (relative to the German university system) were less likely to suffer from this bias, especially if they were themselves critical of German academic philosophers.