checked my own 1851 Navy Conversion made by Uberti. Barrel is stamped thus: "38 Colt & S.W.Spl.". My bore was slugged at .356 and I shoot a .357 dia, 158gr. RNFP bullet. I will admit I've recently experienced severe lead fouling but I believe (with the helpf this forum) that issue is more related to lubing and powder measure than bullet to bore size.
The bores on modern reproductions of factory conversions are much smaller than those for cap & ball, and they will accommodate modern factory ammunition to include smokeless powders. If reloading, keep in mind these are still revolvers of Open Top design so you don't want to go crazy with the amount of smokeless powder you put in a cartridge. Personally I just reload with black powder as was done with the originals.