I've looked in different places and have come up with some interesting reading.
The China Inland Mission, now the Overseas Missionary Fellowship, which had a missionary, James Cameron, in Tibet as early as 1877. They eventually withdrew when the government became aggressive in 1953. Obviously, it's hard to get information after that.
Apparently, though, in the early 20th century, significant numbers of the Lisu, Yi, and Nu peoples became Christian. Some missionaries of note were Isobel Miller Kuhn (1901-1957), Alfred James Broomhall (1911-1994), Joan Margaret Wales (1916-). I'm not sure who they were affiliated with. Their work seems to have taken place mostly on the eastern border.
I definitely can't find anybody who made converts in Lhasa or thereabouts. Fascinating stuff. I had no idea there were missionaries anywhere in that area before the 20th century.