Awesome question. I love this kind of thing.
Wikipedia of course has a a page on near-Earth supernovae, and the first body paragraph briefly examines the effects of life on Earth. The only really important thing there is just what you said - that the gamma rays from a supernova could deplete the ozone layer. Not too specific. But look further down the page. In this section, it says
Recent estimates predict that a Type II supernova would have to be closer than eight parsecs (26 light-years) to destroy half of the Earth's ozone layer.
I checked the referenced estimates on the arXiv pre-print used. It starts off with
Estimates made in the 1970’s indicated that a supernova occurring within
tens of parsecs of Earth could have significant effects on the ozone layer.
They then say that they now have improved tools; later on they say
We find that for the combined ozone depletion from these
effects roughly to double the “biologically active” UV flux received at the surface
of the Earth, the supernova must occur at <∼ 8 pc.
which they restate later as
Summing our gamma-ray and cosmic ray depletions for DSN = 10 pc, and taking into
account that our adopted energy is larger than that found in the latest SN study mentioned
earlier, we obtain a fiducial “critical distance” to significantly disrupt ozone of Dcrit ≃ 8 pc
for a SN with a total gamma-ray energy ∼ 1.8 × 1047 erg.
And in the conclusion, we again find
finding is that a core-collapse SN would need to be situated approximately 8 pc away to
produce a combined ozone depletion from both gamma-rays and cosmic rays of ∼ 47%,
which would roughly double the globally-averaged, biologically active UV reaching the
Not a huge difference at first, but we'd see some pretty bad effects after a while - say, a day or two.
Looking here, we find
A supernova explosion
of the order of 10 pc away could be expected every few hundred million years, and could
destroy the ozone layer for hundreds of years, letting in potentially lethal solar ultraviolet radiation.