Would postwar conditions have prevented Britain from joining another battleship race if naval treaties failed?
Short answerNo. Britain would build if the United States and Japan did, but the race would probably emphasize ship quality and size rather than the numerical rivalry seen against Germany.
NotesBritain would continue building battleships if other powers did so. The Anglo-German contest before World War I was an explicitly numerical competition. A 1920s race with the United States and Japan is expected to concentrate more on technology and the size and quality of individual ships. In that counterfactual, Britain would probably build four G3 battlecruisers and four N3 battleships.