You did not specify which 1860 conversion and which caliber you have but...assuming it is a factory made Uberti RM or R2 their groove dia (factory specs) is .429 in .44 Colt, and .450 in .45 Colt. The corresponding dia of the .44 percussion bore can be between .452 and .458. For safety reasons frame, cylinder and barrel of the conversions are beefed up compared to the C&B model, i. e. you might be able to get the C&B barrel on the arbor of the conversion, but it would not align with the conversion chambers, the 2 frame pins will be off, too.
If your conversion was made from an 1860 C&B pistol with a lined barrel for .44 heel type bullets you might be able to fit a C&B barrel of the same manufacturer to it.
Long Johns Wolf