IIRC, part of the British motivation for intervening as they did, was that it was already "not pretty." However, I think their real motivations were much more practical.
First off, there was the Suez Canal, a primary purpose of which was strategic and colonial, in the sense of shortening Britain's route to India and its colonies on the Indian Ocean side of Africa. They had a very serious interest in ensuring that the canal zone was not endangered by whoever came out atop the "local sorting" process.
Second, Middle-Eastern oil had already become a precious commodity, and again, the British wanted to make sure that their supply of oil wasn't endangered by the wrong folks.
It's interesting to note that there was a significant political divide in Britain over which factions to support. Churchill was among those who favored the Hashemite dynasty (from which Jordanian royalty is descended). Others favored what eventually became the current crop of Saudi royalty. It's difficult to say who was right ... either way, the divergence between Arab and British/western interests was difficult to control.
I would back Churchill. The Jordanians are much more stable (in relative terms of course) than the Al Saud family in Arabia. The Wahabi extract of Islam that is practiced on the peninsula is rather archaic.