It is not fuel, more of a boost , to increase the efficiency of the fission bomb part.
In terms of conventional bombs, HE would be too dangerous to carry around and store if it readily exploded so it is designed to be resistant to heat or damage.
Take some plastic explosive and hit it with a hammer or get a lighter and set it on fire and it wont explode... or it shouldn't... it will burn though...
To get it to explode you need another explosion, so for plastic explosive to blow up you need an explosion to make it explode.
That explosion is called a primary charge or low explosive... a trigger... or a fuse.
With low explosive even a spark will set it off so blackpowder can be set off with a simple burning fuse or spark. A small Black powder charge could be used to set off a much bigger more powerful HE charge... the black powder charge wont be huge... it is not there to create the damage... just to start the HE exploding and it will do the damage.
It is the same with a hydrogen bomb... hydrogen atoms wont fuse together except at extremely high temperatures and high pressures.... it only happens naturally inside a star. To create conditions like those inside a star heavy elements of Uranium or Plutonium are split in a runaway nuclear (not chemical) reaction. The issue with Fission is that you create an explosion by having enriched atoms of the material of a certain mass or volume. Once you have that much material there then boom the reaction becomes a runaway reaction and boom.
Obviously if you need 1.5kgs of heavy elements to reach critical mass then you can't store 1.5kgs of that material in one place or boom it will react.
If you want a bigger explosion you need more mass, but the critical mass remains the same you can't just scale it up.
The minimum bomb size assuming 1.5kgs is critical mass could be two 750 gramme chunks of the material that are slammed together to form critical mass of 1.5kgs and boom. You could scale such a design up by making each of the two chunks heavier... but as they get closer to critical mass they get hotter... two 1.2kg masses wont go boom because that it not critical mass but it might be enough mass to generate temperatures of thousands of degrees C that melts the nose cone of the missile and has liquid uranium or plutonium dribbling down through it.
The easiest solution is a huge ball that has wedges like a pizza except instead of a flat disc with 6 or 8 pizza slices it is a 3D sphere with perhaps 50 segments that can all be pushed from the outside of the sphere into the centre to form a solid sphere of the material... each piece might be 100 grammes each but the 50 pieces added together make up a sphere of 5,000 grammes of 5kgs which will definitely explode.
100 gramme sections wont be big enough for them to get too hot though many bombs had air conditioning systems to keep the elements cool so they didn't damage parts of the weapon.
The simplest nuke was called a gun barrel nuke and had a sphere of material with a section missing. That section was at the other end of the bomb and mounted inside a gun barrel pointed at the main mass of material with a HE charge behind it. When the bomb went off the HE charge blew the missing piece of the sphere down the barrel and into the sphere... its mass together with the mass of the ball created a sphere of critical mass and boom.
To make it bigger and more powerful they needed more pieces... but if the pieces got too big and heavy they would get too hot... if they got to hot they might damage the HE or mechanism to throw the pieces together which might fail and lead to the bomb not exploding properly... that is why they needed to test them.
Imagine a ball like the one I mention above split up into 50 pieces that when slammed together make a ball of 5kgs mass... if critical mass for the material is 1.5kgs then if half those segments are slow and 2.5kgs of the sphere reach each other first then they might blow up and just vapourise the other 2.5kgs of heavy metals so the boom will only be half as big as you thought it would be.
Another design could be to use HE to increase the pressure inside the warhead so it acts like it is critical mass even though it is not and boom but you get a smaller boom than you would if you put two pieces together that reached the correct weight.
In their experiments to create fusion reactors one of the problems scientists are coming up against is that inside a star it is hot but it is also under enormous pressure... so while fusion in our sun might be taking place at 5 million degrees C to do the same here on earth you might need temperatures of 15 or 20 million degrees because the pressure inside the reactor is nothing like the pressure inside a star.