What follows is all public knowledge to physicists, with whom it used to be a popular lunch time topic. The only exception I have found to this is in the dining room of the Trinity Beverage Company in Los Alamos. While dining on one of their delectable “Fat Man” or “Little Boy” burgers, I got dirty looks when I gave a high school level explanation to my companion.

• Miniaturized nuclear weapons have two branches, the artillery shell/suitcase device, and the transportable device.. The shell/suitcase is optimized for minimum volume. The transportable device is optimized for minimum weight, and is very similar to the warhead problem.
• The most simple form of the shell/suitcase, a uranium core in two pieces, weighs a lot, is easy to make, and has a skinny shape. It can fit in a suitcase. At the very low end of yield, it might be luggable  by a strong man without breaking his hand. Modern roller luggage takes care of this.
• A more advanced form of the shell/suitcase is the linear implosion device, which is similar to the device described with the next bullet, but is thinner.
• The  guts of a  transportable device optimized for weight  is a sphere of some diameter containing a unitary plutonium core, and ancillary gadgetry. The external package is typically a can.  It could fit in a very extended backpack, but not in a suitcase.  It can be “boosted” with tritium, increasing the yield, without significantly increasing the weight. The U.S. luggable, the SADM, is pictured here. It’s  a little bulky to roll through Penn Station. I would hope someone would notice.
• At the level of North Korea’s effort, the transportable device is almost identical to a package for a missile.

The development of a miniaturized warhead has two parts:

• The physics. For the top-tier powers, the design of a new warhead can now be done entirely by supercomputer. The extensive experiments of the Manhattan project, and those that followed in the 50’s and 60’s,  were needed to check the then primitive theory, and fill in the gaps.
• Building it. This involves the actual making of parts and piecing them together, skills of  metallurgy, chemistry, fabrication, and electronics. For the transportable device, using plutonium, all these problems are hard.
• Hydrogen bombs are not part of this discussion.

The path of proliferation suggests that the physics knowledge was not independently developed. Every proliferation derived some information from Manhattan.  Additional information was obtained by expat and exchange scientists, and by theft. Because the breadth of the industrial base required to make a nuke is so broad, bits and pieces of the technology are widely distributed. Even when such information is classified, the extent of distribution resulted in ineffective protection. Making a bomb is largely a part of industrial sleuthing. Quoting the NY Times,

It required more than three decades, a global network of theft and espionage, and uncounted millions for Pakistan...it could not have happened without smuggled Chinese technology and contradictory shifts in American policy...

In case you’re blocked by the paywall, you can get something from the Huffington Post: Who Created Pakistan’s Nuclear Arsenal?

Open source suggests that the main stream of proliferation (with side channels and additions) was China–>Pakistan–>North Korea.  The side channel is typified by (NY TImes)

The United States provided Pakistani nuclear scientists with technical training from the 1950's into the 1970's. And it turned a blind eye to the nuclear weapons program in the 1980's, ...

The tinier the weapon, the more distant from the knowledge that became quasi-public after the Manhattan project. De novo development  is difficult for a primitive country.  Did  proliferation proceed identically for Pakistan and North Korea? Was it  by walk-and-talk and paper plans, or was there a specific example?

To be continued shortly.