I can see an Anglo-Italian contretemps over something in the Med leading to an agreement (though perhaps not a full-blown alliance) with A-H, so long as France is not an option. This also removes from Britain the obligation of engaging an army on the continent. If the likely battleground is the Italian peninsular, and Italy is unsupported by the other two power blocs, it would be likely that the RN would do the work at sea, and the KuK Armee the fighting ashore. This neatly removes Britain from experiencing any major changes to the general relationship between her army and navy.
France and Russia would be as IOTL in response to the FPW and that Three Emperors' League falling apart again. Then one has to engineer bad feeling between one or both of them and the British. This would be easier if France had a larger population (or Germany a smaller one), since she could then stand on a more equal footing in a prospective rematch.
Germany and the Ottomans is a bit more complicated. If Germany wished to interfere with the movements of the British in the Med, then the Italians would be a more natural choice, since they are less confined geographically. So it should be something else. Perhaps oil? They did have good links IOTL anyway, with the head of I Corps being a German officer (IIRC), and the Berlin-Baghdad Railway, and so on.