The link provided by @T.E.D. combined with the observations of Major Charles Sweeney that the height of the flash, off the cirrus clouds over Nagasaki, was "at least 6 miles (or 31,000 feet) [above the explosion]" giving a refraction-free visibility for such a reflected flash as about 230 miles or 370 km.
The underside of the great clouds over Nagasaki was amber-tinted, as though reflecting the conflagaration at least 6 miles below. Beneath the top cloud mass, white in colour, there gradually climbed a turbulent pillar of black smoke and dust which emitted a second fireball less vivid than the first.
distance in miles =~ sqrt( 7 * height-in-feet / 4) = sqrt(7 * 31,000 / 4)
=~ 230 miles ~ 380 km
However, as noted further down in the link provided by T.E.D.:
Unfortunately, the refraction varies considerably from day to day, and
from one place to another. It is particularly variable over water:
because of the high heat capacity of water, the air is nearly always
at a different temperature from that of the water, so there is a
thermal boundary layer, in which the temperature gradient is far from
In conditions that produce superior mirages, there are inversion layers in which the ray curvature exceeds that of the Earth. Then, in principle, you can see infinitely far — there really is no horizon.
However, there is one situation in which objects can be made out at great distances: when they are silhouetted against a bright background, such as the setting Sun, or (just after sunset) a bright twilit sky.
What's the record for visibility without help from the silhouetting effect? I think that might belong to the report of the expedition led by Korzenewsky (1923), who reported seeing snow-capped peaks of a mountain range 750 km away. Conditions were perfect: the lower atmosphere was in shadow at sunset; the peaks were quite high (4650 meters, or over 15,000 feet); they were covered with white snow, increasing their visibility; and there must also have been considerable looming to bring these distant features above the observers' horizon.
This observation by Korzenewsky is very similar in circumstances to the reported sighting from Shanghai, so I would not rule it out completely without further investigation of the meteorological conditions that day. It was late summer, with clear calm conditions (breeze < 5 mph at ground zero), which is typical of mirage conditions.
This link goes into the detailed meteorological conditions necessary for a long-distance superior mirage, such as the reported observation must have been. The required temperature inversion could easily have existed in late morning in early August over the cooler waters of the Sea of Japan.
Sweeney can accurately say that "the underside of the great clouds over Nagasaki" were at least 6 miles high, because he himself was at 30,000 feet for the bombing run and aftermath.