If I may... I worked as an "old timer" at Williamsburg and assisted Jon Laubach in forging swamped barrels. His son Chris is the lock maker who participates on this site (or did??)...
The barrels were intentionally forged to have taper. A very basic description of this is the taper was started by creating a "flat" iron skelp that tapered in thickness and width. The skelp was then heated and rounded into a trough by forging on a swag block. Next the edges are wrapped, starting in the middle of the length, and forge welded lengthwise around a mandrel*. It takes many heats in the forge and lots of hammering to do this! The resulting barrel is thus by design thicker at the breech and tapers towards the muzzle.
The flared muzzle is created by "jumping" the barrel. Jumping involves heating the barrel and driving it into the face of the anvil to expand and thicken the barrel near the muzzle. This step too requires multiple heats and you have to maintain a hole through the barrel. If you close up the hole through carelessness you cannot get a reamer through it... barrel drilling wasn't possible in the 18th century so you may have scrap with which to make a new skelp.
The barrel taper is made more uniform and straighter through heats and hammering and then the flats are forged... again using a mandrel near the ends to preserve the hole. Later the flats are filed...
*The mandrel is a steel tapered rod that is used to maintain a hole for future reaming... it will not get welded into the barrel as it is never brought to welding temperature... it only gains heat from being inserted into the hot skelp during welding.
There are a LOT more fine details involved in the forging, dimensioning and quality control but this description gives you a good idea how it was done. A good eyecrometer is crucial throughout the process as are some gauges...
Note that there is a barrel forging video featuring Jon and Chris floating around. It's title "Forging a flintlock rifle barrel with Jon Laubach." I do not recall who produced and distributed it...