I won't disagree but perspective is important:
The chances of experiencing adverse health effects from radiation are slim for both nuclear and coal-fired power plants—they're just somewhat higher for the coal ones.
"You're talking about one chance in a billion for nuclear power plants," Christensen says. "And it's one in 10 million to one in a hundred million for coal plants."
and
McBride and his co-authors estimated that individuals living near coal-fired installations are exposed to a maximum of 1.9 millirems of fly ash radiation yearly.
To put these numbers in perspective, the average person encounters 360 millirems of annual "background radiation" from natural and man-made sources, including substances in Earth's crust, cosmic rays, residue from nuclear tests and smoke detectors.
( ^^ above quotes from above comments link [1] )
So, bugger all in both cases particularly when compared to normal background radiation exposure.
> digging up a whole lot of coal (particularly in Australia, which also has a whole lot of uranium)
Australia is kind of big (the same land area essentially as the contiguous United States of America) and I don't see the major uranium deposits in Australia being especially close to the massive coal beds.
For comparison, the USGS database on Coal quality states [2][3]
In the majority of samples, concentrations of uranium fall in the range from slightly below 1 to 4 parts per million (ppm). Similar uranium concentrations are found in a variety of common rocks and soils, as indicated in figure 2. Coals with more than 20 ppm uranium are rare in the United States. Thorium concentrations in coal fall within a similar 1–4 ppm range, compared to an average crustal abundance of approximately 10 ppm. Coals with more than 20 ppm thorium are extremely rare.
and Australia's CSIRO has it (tables 5 and 6 and end of PDF doc) [4] that Australian coals typically have 1ppm U and 3.5 ppm Th.
These figures for coal are all less than typical crustal concentrations.
The below link [5] is a throw in that doesn't address U-Th-K in Australian coal but does address the major properties of Australian thermal coals.
[1] https://www.scientificamerican.com/article/coal-ash-is-more-...
[2] https://pubs.usgs.gov/fs/1997/fs163-97/FS-163-97.html
[3] https://ncrdspublic.er.usgs.gov/coalqual/
[4] https://publications.csiro.au/rpr/download?pid=csiro:EP11424...
[5] https://minerals.org.au/wp-content/uploads/2022/12/Best-in-C...