Were cancer treatments developed using knowledge obtained in Nazi Germany using practices which would have been considered unethical in liberal democracies of the same era?
I only have one source so far (unfortunately), but it does give quite a bit of information. It's the aforementioned NY Times book review, accessible here. What is says is amazing: German scientists (not necessarily Nazis) were the first to do research on many different facets of cancer. Much of this was pre-Third-Reich, in the glory days of late 19th-century Germany.
However, despite what the author says, the evidence he gives seems to suggest that Germany was at the forefront of cancer research before the height of the Nazi regime. He cites numerous studies in the early 1930s and 1920s, which were before Hitler truly reached power and certainly before the infamous Nazi medical experiments. Only two sections seem to indicate that you're correct:
German scientists in the mid-1930s elaborated on this scientific base. The Reich Anticancer Committee (Reichsausschuss für Krebsbekämpfung) established in 1931 was enlarged, and an ambitious new journal, the Monatsschrift für Krebsbekämpfung (Monthly journal for the struggle against cancer), published by the notoriously antisemitic J. F. Lehmann publishing house, was launched in 1933 to coordinate the anticancer effort.
This supports your position. Continuing on,
More than a thousand medical doctoral theses explored cancer in one form or another in the twelve years of Nazi rule; only diseases of the blood attracted more attention. Cancer registries were established, including the first German registries to record cancer morbidity (incidence) and not just mortality (deaths).
The 1000+ theses could be a coincidence, and may have not been influenced by Nazi ideology. Finally,
Efforts were made to strengthen prevention-oriented public health measures, including occupational safeguards, laws against the adulteration of food and drugs, bans on smoking, and programs to reduce the use of cancer-causing cosmetics, to name only a few.
Technically, this would have been a Nazi measure, but it may not have been motivated by Nazi ideology.
The second section is more direct:
A broad range of Nazi officials—from Gauleiter Julius Streicher to Reich Health Führer Leonardo Conti—championed a renewed focus on preventive medicine. The Monatsschrift für Krebsbekämpfung adopted the motto: "The earlier a tumor is treated, the better the likelihood of a cure." The Zeitschrift für Krebsforschung announced that "prevention is the best therapy." The ideology of prevention merged with the ideology of "one for all and all for one" (Gemeinnutz geht vor Eigennutz) that was yet another hallmark of Nazi thought: as one anti-tobacco activist put it, nicotine damages not just the individual but the population as a whole.
This all comes after a lengthy section on Erwin Like, a prominent oncologist who vocally supported the Nazis. Whether or not his research was Nazi-motivated may be contested. Overall, though, the Nazis enacted some cancer treatment and prevention measures, but not a whole lot. Although I only have the one source so far.