This is my first post, and I'm therefore a bit uncertain when to post an answer. But it's been about a month, discussion is winding down, and I am pressed on with the writing. So I thought I might share my answer:
Before providing it allow me to first say, thank you. This has been a very interesting experience and quite engaging. I'm happy to say that the following ideas would not have been possible without this discussion.
It strikes me that the moment to stop answering the question isn't when we've built a fully valid proposition, but rather when the needs of the story have been satisfied. In this case, my story is set a few hundred years into the future, and the disruptive event (the Rise) look place about 100 years from now. It seems quite reasonable as well as dramatically interesting for the characters themselves to wonder whether or not the earth rose around the buildings or whether the buildings fell into the earth (as per Inbar Rose's reflections above). If two cultures develop legends and lore around those alternative readings of the physical world, it becomes a dramatic rich point for story development. In my case, I think this works quite well.
That does not relieve the burden, however, for me to create a plausible set of conditions for the world to be in such a state (i.e. my original question). It does, however, relieve me from having to articulate it in the body of the story itself (which is not an evasion but a deference to the storytelling).
So here's what I'm thinking which will result in the world I am building but need not be mentioned in the story:
Massive solar activity disrupts communications at a politically sensitive time, thereby undermining international cooperation because conspiracy and suspicion about "what happened" overshadows scientific evidence; There is agreement on the solar flares, but beliefs hold that some countries or actors took advantage of that for gain. Global warming massively increases as cooperation ends and regional wars begin using weapons that I fear will be more in use 100 years from now (if multilateral cooperation cannot halt the proliferation) including biological and chemical weapons coupled with cyber attacks on infrastructure. Man-made pandemics with cascading consequences become very plausible. That takes care of most of the humans and many of the animals.
Deforestation does indeed take place with global warming and the solar activity and the (previous) man-made contributions results in "global weirding" (a term now used to explained the intensification of locally strange weather as a function of globally warmer weather). At that point, I feel I can reasonably stop because I can imagine (with your help above) a set of plausible conditions where a city (not all cities) face the dramatic "rise" I mentioned.
As it happens, I do think the earth will have risen (not the buildings having fallen), but the debate is of greater value to my story than the immediate answer.
A few final thoughts on details for those of you kind enough to share your ideas:
I think earthquake protection and higher building standards are more likely as buildings go up from now (2017) to the time of the event in at-risk areas (LA, Tokyo, Rome, Tehran, etc.). So it seems that the most modern of buildings by that time will be those most likely to survive. Consequently, most of the older buildings will indeed have fallen. So the Old World poking through might mean six or seven buildings, not 50 or 60.
Deforestation, desertification, changing of the quality of top soil, the movement of earth (i.e. soil) from nearby mountains down into the city's valley — all these matters could conjoin to create a plausible (local) scenario.
I agree that changes to ocean currents, desalination and other matters can be viewed as primary causes in the wider model.
Thank you all very much! This is connected to what I hope will be my fourth novel and I will announce its title if/when it sees the light of day.