damoc:
Originally, barrels were forge-welded from strips ('skelps') of iron around a mandrel slightly under the desired bore size.
Most such barrels were made-up of a number of short skelps butt-welded together and wrap-welded around the mandrel, though some barrel makers made a long, flat skelp and 'folded' it around the mandrel which resulted in a finished tube with one longitudinal seam
The barrel forger took some care to keep the tube as straight as possible, and of as near the final outside shape and dimension as could be done.
Because the mandrel tended to stick in the bore if too long a weld was attempted, it had to be driven out and re-inserted about every inch of the welded length.
When the tube was completely welded, the bore was reamed-up to final diameter and finish using the short bit (a twisted, square reamer on a long shank), followed by the long bit (a square, steel cutting bit several inches long, attached to a rod long enough to completely pass through the tube) - the bit was arranged to be raised for successive passes by inserting shims between the bit and the rod until the final desired bore diameter and finish was achieved. The long bit actually can produce a very good bore, both for consistency of diameter and finish, but is very labor intensive. It is also true that the bores of many antique barrels are not truly straight, and never were.
Finish-reamed blanks were usually straightened before finishing the exterior of the barrel, or rifling it - this was done by eye, looking through the bore at a straight line or edge, rotating the barrel, noting where and how the shadow cast down the bore appeared to deviate from a straight line, and bending the barrel with a press or hammer and anvil until it was visibly straight. There were other methods, too, but this is the best, and is still used, though most modern barrel makers do not need to straighten their barrels, and it is better not to do so, if not necessary.
There are still practitioners of traditional arms making who forge-weld barrels, ream and rifle them with traditional tools - there is a very good video of the entire process of making an American longrifle at Colonial Williamsburg in Wallace Gusler's shop, later operated by his apprentice, Herschel House - the video is 'Gunsmith of Williamsburg', and can probably be viewed on-line.
Also, there is an excellent article on the processes and tools in the 'Foxfire 5' book, one of a series on traditional arts and crafts.
Hope this helps.
PRD1 - mhb - Mike