Depth may not be your friend, but number of layers is. When the shockwave hits the surface of the water, much of it is reflected, but some of the shockwave is transmitted. How much? It turns out they wrote a book about it:
As discussed by Henderson et al. (1990), when a blast wave first contacts a water sheet, a large portion of the incident energy is reflected back towards the source due to the impedance mismatch at the air-water interface. The remaining energy, which is transmitted through the water sheet, forms the observed weak shock wave and the resulting initial pressure rise. For the experimental conditions considered here, the initial blast overpressure is mitigated by as much as 80%. This indicates that a water sheet may be an effective emergency blast mitigant.
In this case, the explosive was 48.4g of PETN (page 14), but I don't have any reason to expect any one shockwave to behave differently from another.
So how far do you need to be? Well, they haven't written a book about that, but they did write a text file about it. I'm getting a little nervous just how much attention this problem has gotten...
Blast effects of nuclear bomb (this is for an 80 col. printout)
All distances to effect in miles. Note: airburst distances in ( )
Airburst for optimum damage for that effect, since the height of airburst
changes these figures represent worst case. See example for fixed height
MT 1psi 1.5 3 6 10 30 overpressure
0.2 4(7.5) 3(6) 2(3.4) 1.3(1.8) 1(1.2) .55(.6)
0.6 6(11) 4.5(9) 2.8(5) 1.8(2.6) 1.4(1.7) .8(.9)
1.0 7(13) 5.5(10.5) 3.3(6) 2.2(3.2) 1.6(2) .95(1.05)
5 12(23) 9(18) 5.5(10) 3.7(5.5) 2.7(3.5) 1.6(1.8)
20 19(35) 14(28) 9(16) 6(8.5) 4.3(5.5) 2.5(3.4)
(Update note: the 5 & 20 Megaton bombs only existed in old Soviet Bear and
Bison class bombers and have been replaced with more modern 1 Megaton bombs.
The old US Titan missiles with their 9 Megaton bombs were scraped during late
1987 and early 1988)
Examples of damage (from SURVIVING DOOMSDAY -Clayton, from tables in THE
EFFECTS OF NUCLEAR WEAPONS 1977 -Gladstone)
0.5 psi Private airplanes damaged but flyable, windows have light damage
1.0 psi Windows heavily damaged, wood frame houses lightly damaged
1.75 psi Some, but not all, glass shards capable of penetrating abdominal wall.
2 psi Human body thrown hard enough to cause incapcitating injuries if standing
3 psi Human body thrown hard enough to cause 1% fatalities if standing up.
4 psi Forest road impassable due to fallen trees.
5 psi Wood frame house collapse, 1% of eardrums rupture (in the elderly)
6 psi Human body thrown hard enough to cause 99% fatalities
From this chart, 6 psi is enough for a 99% fatality rate. You're in a submersable, so that's 6psi in the air of the sub. The reflection off the surface means you need 5x that shockwave pressure (reciprocal of 80%), which is 30psi suggesting .95 miles of distance is needed. However, there's more to this, because you have two impedence mismatches. One from air to water, and one back from water to air. These effects stack, meaning you need 25x more pressure at the surface to affect you. Thus, you would need a 150psi shockwave hitting the water to reach the essential 6psi. If we look at the table, that pressure is remarkably hard to hit. Obviously the table doesn't go that far, but I'd be tempted to guess its around 0.1mi, or about 500 feet. As a point of comparison, Little Boy, the bomb dropped on Hiroshima, was detonated at 1,968 ± 50 feet. I won't play CONOPS games with nukes, but I'm guessing that by the time the shockwave hits the water's surface, it's gone far enough to be below the magic 150psi level.
How about radiation? Piece of cake. From our sister site:
We know from the nuclear power industry that spent fuel storage pools
are pretty safe places to be around, radiation-wise. They're actually
safe to swim in, to a point, because they're serviced routinely by
human divers. They just can't get too close to the spent fuel.
We use these pools for short-term storage because water is a really
good radiation shield. How good? Well, according to a report on the
topic prepared for the DoE back in 1977, a layer of water 7
centimeters thick reduces the ionizing radiation (rays and particles)
transmitted through it by half (the remainder is captured or moderated
to non-ionizing energy levels, mainly heat).
Considering that you can hold your breath and dive to 700cm in a pinch, radiation is going to be a non issue for any submersible.