no - all the equipment in the linked video is hopelessly obsolete. not too long after the shot in question (upshot-knothole grable) the us developed 155mm/203mm afaps (artillery-fired atomic projectiles); these could be used by ordinary field artillery. these were rather inefficient weapons, in terms of their yield-to-weight ratio; the diameter of an artillery shell is apparently too small to fit an implosion-type assembly system of the regular type; it's necessary to use a different, much less efficient system. leaving that aside, not least because the range of artillery shells is rather short, it's better to deliver tactical nukes with missiles - ballistic or cruise - or with combat aircraft.
United States