Another hint might be the internal magazine box.
Original Jeffery .404 rifles that we have seen here in Australia's Northern Territory have seriously cut-back mag boxes to allow closing the bolt over 3 rounds down. There may be no metal sides at all to the internal box, just timber.
Worth a look anyway....

And that extreme taper to the barrel half way down is quite characteristic of Jeffery .404 rifles in my limited experience, however British proof marks on the side of the Nock's Form will add weight to the assertion.

Nice rifle, providing the bore cleans up well enough to shoot. I too would be very happy to be planning a tidy-up of that marvelous old campaigner.