Steve, already agree with you that 15625 was originally an 8-bore shotgun and have posted two Reilly's close o 15625 to that purpose.
But that gun
25363 has been in my database for a couple of years. It was serial number in mid-1883 - then repurposed from a rifle to a shotgun post 1903 and rebarreled in Birmingham.
https://www.rockislandauction.com/detail/80/288/engraved-edward-m-reilly-co-8-bore-hammer-shotgun25363 - E.M. Reilly & Co., 295 Oxford Street, London*. 8 bore, SxS Shotgun*. U-L hammer gun., 28” steel barrels (*rebarreled/repurposed from a large bore rifle after 1903)
Now here's something I mentioned above that needs to be contemplated. There is that authentic Jules Julia sold 4 bore wild fowler above
18860. It's all original and in amazing condition...nickel plated and all for salt water - dated 1874 per my chart, 42" barrels... But it's proofed for
6 bore black powder. with 4" chambers and no reproof indicated and sold in UK. So WTF?
-- You've said there are no 6 bore cartridges from that era.
-- Donald Dallas has said he's never seen a UK cartridge for shotguns longer than 2 1/2" in the 19th century.
-- The UK guys say a 4 bore, if it were rebored, had to be reproofed.
So why the "6" on a 4 bore? Sold without a whimper by a noted UK auction house?
Something's amiss...I need to be educated.
Morphy has decided to erase all James Julia History. 18860 was here:
http://jamesdjulia.com/item/1363-396/Fortunately I've kept the original advertisement:
MASSIVE FOUR BORE E. M. REILLY HAMMER WATERFOWLING SHOTGUN IN HIGH ORIGINAL CONDITION.
SN 18860. Cal. 4 bore. 4” Chambers. 42” Dovetailed stub Damascus bbls are engraved “E. M. Reilly & Co Oxford Street London & Rue Scribe. Paris.” on relatively narrow concave top rib. Bottoms of bbls are stamped with London black powder proofs for 6 bore and with SNs. Large nickel-plated Jones underlever action with non-rebounding peninsula back locks have round bodied serpentine hammers and back sliding safeties. Locks have four positions; fired, 8th cock, half cock, and full cock. Safeties engage at 8th cock. Action and locks are completely unadorned except for makers name on the tail of each lock. SN is on trigger guard tang. Lightly streaked and figured European walnut straight grip buttstock measures 14-3/8” over nickel-plated buttplate, and has classic point pattern checkering with mullered borders at grip, and a vacant silver oval on toe line. Matching splinter forend has shaped steel tip, and attaches to bbls with sliding side bolt through oval escutcheons. It appears this gun was made before chokes were invented. Diameter at muzzles is .952. Drop at heel: 2”, drop at comb: 1-13/16”. Weight: 18 lbs. 12 oz. LOP: 14-3/8”. CONDITION: Excellent, very close to new. Bbls retain nearly all of their Damascus brown with only slight silvering at muzzles, and some overall flecking. All other major metal parts retain nearly all of their orig heavy nickel plating, with only a few scattered knocks and a small bit of peeling at toe of buttplate. Action and trigger guard screws retain most of their orig case hardening color. Minor pins and safeties retain most of their orig fire blue. Stocks have nearly all of their orig hand rubbed oil finish with scattered marks and scratches, checkering very lightly worn and dark. Mechanically excellent, but triggers and locks are somewhat sluggish due to congealed oil. Bores are excellent, with some light scratches. Action is tight. Waterfowlers such as this are seldom found in this condition, because they are usually heavily used in salty environments. A superb condition example like this is a great rarity. 51401-5 MGM170 (20,000-40,000) 
