When I had my blacksmiths shop this is how we did it.
Depending on how 'full' the rivet snap impression is (a rivet head is not a true hemi-sphere) and the thickness of material to be joined. Calculate the length of shank required. To form a rivet head you need 1.5 x dia. of shank, plus 1/8''* length measured from under the head, then add the thickness of material. This will give you a starting point for the trials. Using scrap material of the same thickness 'sett**' your first rivet.
Place one of your 'snaps' in a vise, vertically, and have the other snap handy. You will need a ball-pein hammer and some form of heat. Oxy gear is very good.
Place your rivet into the snap & place the material on top, the hole should be a size that is the same size as the rivet shank, a 'snug fit', or you can go 1/64 over on a small rivet. You will probably need to debur the cut end to get the material on.
Using an oxy torch, start to heat the shank at the bottom, move the flame around the shank, you need to get the shank hot where it is under the surface of the material. Don't burn the shank by having the torch too close. The heat will radiate up the shank, when the end of the shank is hot start playing the torch all around the shank and the entire length until the shank is yellow and has a 'waxy' appearance. Quickly play the heat around the base again, put the torch (safely) and quickly to one side. You need the bottom of the shank a little hotter than the top***.
Using the flat face of the hammer strike the top of the rivet with a few deadblows until the shank is reduced in height by approx 1/3rd to just over half its original height. Quickly, using lighter blows with the ball end roughly shape the 'mushed end' into a hemisphere. Do not keep hitting the mush after the red heat colour has gone. (This could cause the edges to split).
Reheat the end until a bright red colour, place the top snap over the mush and drive the snap down using deadblows, when the snap reaches the material being riveted, stop. Or, if the rivet is too long material will be forced out underneath the end of the snap, (such rivets are called "Jockeys Caps" and will see you earning a DCM).
Judging by what you have you will know whether to add or subtract material from the length of the shaft. (Which you did write down on the workbench didn't you?)
You will hear the rivet cool, (as it shrinks it will pull the materials tight).
* Plus 1/8th is the theory, I've always found it to be too much, however it does depend on how full the snap head is.
** Set in the blacksmiths world is always spelt with two 'T's' eg, Hot Sett, Cold Sett.
*** Heating the bottom of the shank hotter than the top causes the hotter metal to expand and fill the hole before the head forms, this ensures a tight rivet. If the head forms before the hole is filled the material can move laterally and act as a shear.
As far as riveting awkward pieces (like a chassis) we would find a way to support the ‘anvil snap’ by placing solid pieces underneath, even wood blocks with a flat piece of steel on top would work. Or we would make (or weld) the snap to a steel bar of the right length to the floor. If you have a mate to help he could hold a heavy sledge against the end of the anvil snap to absorb the riveters blows. This is a sleeves down and buttoned job, no gloves. Expect some minor sparks to stick to your skin.
