The best known Triumph built for the British army during the Second World War was the 343 cc OHV 3HW single, along with smaller numbers of the 343 cc 3SW and 493 cc 5SW side-valve models. Two side-valve twins were considered in the same displacement, but when the Coventry factory was bombed in 1940, the 343 cc 3TW twin was shelved.

The side-valve 493 cc 5TW was built in 1942 with the bottom end from the prewar Speed Twin and a chain-driven camshaft. The cylinders were cast iron in a single unit, the head was alloy and it had telescopic front forks and a rigid rear end. There was still high demand for the OHV 343 cc 3HW single, though, and the 5TW was shelved until the end of the war.

In 1946, the side-valve twin was revived as a military model, and it became the TRW that would be sold from 1950-65. The TRW received the trials frame from the TR5, a four-speed gearbox, nacelle and Trophy wheels along with military equipment. The TRW kept cast iron cylinders and alloy head with the single Solex carburetor, but had a gear-driven camshaft. It gained coil ignition in 1956, and one example was built with a swing arm, but the rest kept the hard-tail until the end.

Almost 6,000 were built from 1950-65 and pressed into military service and police use all over the British Empire. Eventually, their lack of horsepower prompted their replacement by new models with full suspension based on the OHV single-cylinder Cub or twin-cylinder Tiger.

The OHV 343 cc HW was made after the war for civilian use and surviving Army models were repainted. By the time they were all worn out, Triumph was building a whole new range of OHV twins like the T5 500 cc Trophy, and Tiger and 650 cc T110 Tiger and T6 Thunderbird. Even so, the TRW slogged on. Because so many were built, surviving TRW models are not that hard to find, and they are relatively to equip and present one like a service bike.