Very nice find. I'm pretty sure Jeffrey was not a gunmaker, but rather a retailer, though he did hold patents for various accessories & developed a number of proprietary cartridges. Here's some further info I found on the 'net:
William Jackman Jeffery may have started his career in the gun trade as a salesman for Cogswell & Harrison in the late 1880s. In addition, he also worked for P. Webley & Son, but he set up on his own account in 1887 at 60 Queen Victoria Street, in the City of London. In 1891 the business became known as W.J. Jeffery & Co.
Jeffery was interested in rifles and was granted several patents for shooting accessories, including a bore mirror, a telescopic sight and a sight protector. He also developed a number of proprietary rifle cartridges which bear his name. He is best remembered for the mighty .600 nitro express but also designed the .450 No. 2, the .404 Jeffery, the .433 rimless and flanged cartridges, the .333, .280 and .255 rook rifle.
The .333 cartridge proved effective in the early months of World War I by defeating the steel mantlet plates used in the trenches by German snipers.
W. J. Jeffery died in 1907 and the firm was taken over by his brother Charles, who died in 1920. His nephew, F. Jeffery Pierce, took over and in 1927 the firm moved to Golden Square, where it remained until 1955. It was bought in 1957 by Westley Richards and then by Holland & Holland in 1959.
So, according to the business address on your rifle, it seems it was made in 1891 at the earliest.....which does seem a little late for a Martini intended for military-style competition (as the Lee Enfield was the service rifle by then), but perhaps not. My Westley Richards ZAR was made in 1898, & there were a number of contracts for .577/.450 M-Hs into the early 20th century so the rifle/caliber were not totally obsolete. Both Westley Richards and Holland & Holland are still in business, so they could have access to Jeffrey's records.