Fashoda and the Niger Crisis were the height of Anglo-French colonial hostility, and IMHO you'd need to have this explode, even if only into a short sharp shooting war that the politicians cool down. That would probably be enough to antagonise the French into an unfriendly position during the Boer War, and even with Edward VII's accession the idea of a British-French alliance won't be looked on with much favour.
There was talk 1900, 1901 of an Anglo-German alliance with some politicians (Joseph Chamberlain?) favouring it, so add in a shooting war against the French and it might have had more legs.
Given both of these, Britain is going to look much less favourably on Russia and France during the Russo-Japanese War and probably protest much more strongly the use of French bases in Madagascar and Indo-China by Rozhestvensky and his fleet. It would be ironic if this prevents him from reaching the Far East at all, tho!
Internally, there were already changes in the federal empire under way by 1914 - for example Brunswick had been restored to sovereignty after the marriage of the Kaiser's daughter. There was some talk that Hannover might one day be allowed this. And of course the other, constant, question was over the anomalous 'imperial province' of Alsace-Lorraine. If this had been turned into a sovereign state, maybe a duchy under an acceptable ruler (a minor Habsburg?) then it would go some way towards alleviating the concerns of the French minority
Best Regards
Grey Wolf