Best bet for this would be recognition by Britain after the 1904 Younghusband Expedition. IOTL Britain and Tibet agreed a fairly draconian treaty but it was practically unenforceable. In any case, in 1906 Britain came to an agreement with China on Tibet wherein it acknowledged Chinese suzerainty. Seems to me that there's a window in this period where Britain could have created a protectorate over Tibet and, in the process, recognized its independence from China. The biggest impediment would seem to be Britain wanting to do it in the first place. But seeing as how the whole expedition to Tibet was sparked by fears that Russia was plotting to extend its influence over Tibet, that would seem the best route for making Britain create the protectorate.
So how about this: Russia has a bit more involvement in Tibet, maybe an official mission or two makes contact with elements in the Tibetan government. Maybe that happens in combination with Russia using the 1896 agreement with China to push its interests in Central Asia as well as in Manchuria. That would create a driver for Britain to peel Tibet away from China rather than just making sure the Russians are kept out. So, when Britain gets twitchy over percieved Russian moves in Tibet and intervenes as IOTL, the resulting treaty establishes a British protectorate over Tibet. China would complain about it, but there's nothing they can do about it practically. There you go, Britain recognizing Tibet as an independent nation, albeit one under British suzerainty. Be intersting to see what happens to Tibet after Indian independence in this scenario.
