A couple thoughts from the peanut gallery here. Please don't take it as casting shade on your gun, its a British made double rifle, so it isn't all bad.
The gun has been worked over in a number of troubling ways. Take stock in everything
@Red Leg wrote herein.
The gun is the lowest grade double rifle Westley Richards made, and the crudeness of it is significantly below what we call a "gold letter" model from the 189-1910 era guns. Nonetheless, saying its the worst Westley Richards ever made is like saying the Ferrari 308GT is the worst car they made in the past 40 years...it's still a ferrari! The labeling (gold inlay) is really rough...not anything like the steady hand of the several WR gold letter models I've owned which causes me to raise an eyebrow at what I'm seeing.
35,840psi is the proof pressure listed on the proof marks. The 16 tons per Square was BRITISH LONG TONS so please make sure to keep that in mind. So the 3" 500NE has a proof pressure of 40,600 psi so its safe to say that indeed, this is not a "Full, modern, 3" 500 NE" gun. So what is it? Chamber cast is essential to find out if its a 3-1/4" chambered gun. Shooting 3" loads could have caused some fairly serious issues to the gun so getting it inspected is important at this point. Also, VERY CAUTIONARY warning as the bushed strikers look wrong. They are bulging out of the action in an odd way. Please do NOT fire this gun as it is right now. Something is very wrong.
Some data points to try to figure out what it might be based on proof marks. 500 Nitro for Black Express 3" had a service pressure of 11 tons per square, so the 16 tons per square proof pressure would be a possible option. The service pressure of 500NE 3" was 16 tons per square, so that matches the proof pressure of 16 tons per square, ruling that out as an option. (proof pressure for safety testing is higher than working/service pressure) So it isn't for a 3" 500NE with a 570gr bullet and 80 grains of cordite at 2150fps out of a 28" barrel...it has to be something less than that.
I would pay Wesley Richards to run a history and build report on the gun, their notes might say exactly what it was. It could be some 500 3-1/4" light nitro odd-ball using a lighter bullet, many strange things were made. The "BNP" mark for Birmingham Nitro Proof at least indicates its a nitro load of some sort, not black powder only. The proof mark just says ".500" which is an odd mark compared to all that I've seen and owned to date as well. Definitely worth a chamber cast to make sure its not a 500-465, 500-416, 500-something-else.
Update: there is a circa 1957 WR that sold through one of the normal dealers awhile back. Virtually identical to your gun in fit/finish. Even the less than professional gold name compared to the earlier years. It weighed 10lbs 7 ounces. It was a 500-465 NE. I'd weigh your gun to see if that matches yours? The case length was often 3-1/4" and the service pressure of 14 tpsi would match the 16tpsi proof pressure well also. The 500-465 is a 500NE case necked down to hold a .470 caliber bullet. That would all add up nicely to put the pieces of the puzzle together.