All the cool kids are playing with the new Google Ngram viewer, which lets you make graphs of word usage over time. You can search for usage in Chinese, but I’m not sure if they have enough Chinese books to get much out of it. Here is a search for Beijing, Peking, and Peiping.
You can see the steady decline of Peking after Nanjing becomes the capital in 1928, in part I guess because some people are using the new name of Peiping and in part because there is less reason to refer to the city when it is not the capital. (Plus, historical books might keep using Peking. This is based on date of publication. I’m not sure what they do with reprint editions.) The thing that really surprises me is that Beijing gets almost no hits for a very long time. If you fiddle with the controls almost all the early usage of Beijing comes from British English, and not much even there. Peiping dies like a dog after 1949 in British English, but the Americans keep using it for a bit. Whatever sort of English, however, the late 80’s is when Beijing and Pinyin take control.