I have to admit I got a bit ahead of myself in the post below, in which I renamed the nuclear cannon the Verne gun and described some of what you could do with it.  As it stands, the idea would only work for cargoes that could withstand tens of thousands of g's of acceleration---which in practice would amount to fuel, raw iron and a few other simple items like that.  Still valuable to orbit, but a bit limiting.

So, here's a proposal to refine the idea a bit:  the sabot.  In this variation of the Verne gun, you don't try to reach escape velocity.  The blast that sends up the ship only needs to loft it about 100 kilometers---above the atmosphere, but not into orbit.  The bulk of the ship's mass is in fact acceleration padding--a sabot or shell around a more conventional rocket-powered craft.  After an initial acceleration (still on the order of hundreds of g's at least) the sabot separates from the cargo at 100 kilometers, lightening the load and permitting the contained rockets to fire.  This lighter craft then enters orbit under rocket power.

An alternative to rockets would be to catch the ship at the top of its trajectory using an orbiting tether (a huge one, if we're catching tens or hundreds of thousands of tonnes!).  In either case, the acceleration shielding for the initial launch falls back into the ocean and what enters space is pure cargo.

Using a sabot might allow us to launch more fragile cargoes than the straight shot version.  I now doubt that you could launch, for instance, solar power sats without a sabot, though sending up a space elevator would probably still work.

Toby Buckell informs me, by the way, that Niven and Pournelle used the idea of the nuclear cannon in their alien-invasion novel Footfall.  Let's get precedent straight here---as far as I know, they did it first in science fiction.