Russia had to get involved in the Balkans as they had been doing that since the mid 1800s, freeing Orthodox states from the Turks - they felt it was their national goal to reconquer constantinople and that the Austrians stood in their way
The Brits could have sat it out - their empire would have lasted longer
Agree about Russia & Austria-Hungary!
And yes the Brits would have been much much better off sitting on the sidelines offering to mediate.