There isn't one because they're measurements of different things.

With respect to the original standards of the British rifle cartridges, "tons" is a measurement of bolt thrust. The British ignored chamber pressure. They standardized their rifle cartridges in base crusher guns that measured bolt thrust using the Base Copper Unit of Pressure yardstick, with the result expressed in "tons" - 2240 pounds/ton. CUP (Copper Unit of Pressure) and the current BAR/PSI are radial measurements of chamber pressure measured by crusher and piezo electric transducer methods respectively. None of the three are convertible to each other.

No, the Birmingham Proof House doesn't use "tones". The old standards have been out of use for 18 years. Britain is now a member of CIP and their proof laws are now standardized on the CIP mandated chamber pressure measured by piezo electric transducer, expressed in BAR. Most of the old British rifle cartridges have been standardized by CIP, and the old bolt thrust standards have been laboriously converted to current chamber pressure methodology. Since bolt thrust varies with regard to case dimensions, the current standards can be surprising to some. For example, the .450 No. 2 NE, at 13 tons, had the lowest "pressure" of the large, full nitro, flanged cartridges, but it's chamber pressure is far from the lowest.

.450 No. 2 NE = 13 tons. Current CIP MAP = 2800 BAR/40,610 PSI
.470 NE = 14 tons. Current CIP MAP = 2700 BAR/39,160 PSI
.500/.465 NE = 14 tons. Current CIP MAP = 2450 BAR/35,534 PSI

BaseCUP (tons) and CUP are both long obsolete.