Like most (if not all) British commercial arms makers in the 1860s, Barnett's products were sold to agents of both North and South. Perhaps more went to the South - it's hard to say - I have seen more Barnett Enfields with Confederate markings than with Federal.
Barnett's P1853 Enfields are somewhat different from other makers', since there was a variety of lock markings. Most makers used the same marking on their products, but I have seen Barnett P53 Enfields marked in all of these ways:
BARNETT/LONDON, with the BARNETT in italics, with nothing behind the hammer.
Same as above, both in italics
BARNETT/LONDON, BARNETT in italics, normal large crown behind the hammer, with TOWER beneath
Any of the above, both with and without the double line engraved around the edge of the lockplate.
Barnett also made other types of Enfields, including the P1856 cavalry carbine.