I'm actually deeply suspicious about the data used used on TimeAndDate for historical times, and I suspect things were a lot more messy on the ground in China in late 19th to mid-20th century, even with growth of telegraph etc.
I did, however, poke around some postings in Chinese about Jon Skeet's Stack Overflow posting. One of the commenters here, wubotao, pointed to this passage on the Chinese wikipedia entry for the Chinese Time Zone (中國時區)
Emphasis by me on the key line, which may have something to do with it. This is the only entry for 1928 (which as @Henry points out, is right in the midst of the Northern Expedition unification effort).
Basically this says there is a 1928 shift in control from the former central astronomical institution (中央觀象台) to have its duties divided between the new Nanjing government's astronomical research institute (天文研究所) and its meteorology research institute (氣象研究所). While latter two based their almanac on the work of the previous one, they no longer use Beiping (now again called Beijing) standard time for time zone calculations but use Nanjing standard time
It says at 120° but that is not Nanjing but more like halfway between Shanghai (around 121°) and Nanjing (118-119°). It is, however, further east than the old capital Beijing (around 116°ish)
Not entirely clear to me exactly how this worked since I thought there would have to be some standardization with other non-Chinese timezones, but it is all I was able to find with a quick search on the Chinese side of this discussion.
I looked through the December 31, 1927 and January 1, 1928 issue of the Shanghai based Shenbao (申報) newspaper but didn't find any articles warning its readers of the coming time change.