While I served my apprinticeship at the Ohio Village Gunshop in the mid '70s we hand forged several barrels.
We started out with a "skelp" of wrought iron about 40" long by 4" wide by 3/8" thick.
Starting out from the middle of the skelp it is forged into a "U" shape out to one end. Then turned and forged out to the other end. Again starting in the middle the edges are brought together till they slightly overlap. The skelp was then heated and fluxed, placed back in the fire and heated to a white hot heat, it is placed on the anvil and a mandrel inserted into the bore and the edges briskly hammered together to close up the metal and make the weld. This is repeated clear out to one end, you can only weld 1" to 2" with each heat, it is then turned and welded out to the other end. When finished you have a rough forged tube about 44" to 46" long. The barrel blank is placed back in the forge and the rough octagon shape is hammered on the outside and the swamped shape is approxamated. The barrel blank is heated to a bright red from the middle to one and buried in sand or coals and allowed to cool slowly, again the blank is turned and the same thing done to the other end. This anneals the iron in preparation for the boring operation.
The barrel blank is then put on the boreing machine and a small diameter bit is started and slowly turned through out the rough hole that will be the bore. Successively larger diameter and longer cutting surface bits are run through the blank which cleans out the roughness and straightens the bore somewhat at the same time.
After the rough boring operation the barrel blank is "set up" or straightened from any warpage caused by the rough boring operation.
The square fine boring bit it run through the blank and it removes the marks from the rough bits and leaves the bore relatively smooth.
The outside is draw filed or ground to shape and the blank again straightened if needed. If it is a smooth bore it is then breeched, if it is going on a rifle it goes to the rifling bench and then breeched.
A very good description of barrel forging and finishing in the 18th century is given in the book "An Essay on Shooting" originally published in London in 1789 but was available as a reprint recently.
Hope this helps.
Regards, Dave