I will start from the same premise of PcMan's answer
Your cannon has a muzzle velocity of 625m/s
It is firing a continuous stream of 15 kiloton nuclear bombs, at a rate of 150 per minute, for a minute duration.
Your bombs are spaced apart by only 250m.
The first detonation occurs before you stop firing.
Now, consider that a nuke is designed to explode only when a set of carefully designed circumstances is met. As again correctly stated in PcMan's answer, no nuke is designed to functionally survive inside the primary fireball of a nuke, because if it was it won't explode.
This means that all the nukes after the first which happen to be in the primary fireball of the first explosion would simply fizzle and not explode. There is no sympathetic explosion for nukes.
But how big is the fireball of a 15 kTon nuke? Believe it or not, somebody has made a calculator for that!
For 15 kTon that calculator gives a fireball of 130 meters and a duration of 0.7 seconds. Thus, if we assume perfect firing (no spread due to recoil etc. and aim always at the same point) every other nuke will be inside the fireball (you are firing one ever 0.4 seconds). Everything fine then?
Well, consider that they would still be subjected to the thermal radiation and to the EMP. Now, I assume that whoever designs the controlling circuits for nukes knows that they need to protect them from EMP, else MAD would turn into a joke. And the thermal radiation should be manageable, too. When nukes were exploded above ships, the ships were wrecked but not vaporized.
You seem to have then about 75 Hiroshima bombs in close succession. What happens to the target? Since a single bomb on Hiroshima caused a firestorm, tore the city down and made a carnage of the inhabitants, and considering that more bombs won't bring more combustible for the fire nor more buildings to destroy nor more people to kill, the only serious consequence of using this weapon would be an increase of the fallout and of the long term and long distance contamination.