Yes and no. I cannot tell from the picture but the black bezel suggests a mid-1970s gauge. After about 1975 the Triumph speedometer cables "clipped" onto the gauge. Prior to that, the cables attached with an M12x1 threaded nut... just like other British cars of the period. The later cars had cables that used a flange and fork to attach to the Triumph gearbox, the earlier cars had the unique 3/4-26 threaded nut to attach them to the gearbox.
Please note the numbers at the bottom of the gauge face, in particular "1000". That number indicates the gauge is calibrated for 1000 cable turns per mile (TPM). To use it on your Mini you will need to either 1) have the odometer gearing replaced and analog needle recalibrated, or 2) you need to have an inline ratio converter gearbox made up to adapt the Mini's speedometer cable drive speed matched to the 1000 TPM number. The ratio gearbox will probably be less expensive than having the gauge converted. Some will suggest changing the speedometer drive gears in the transmission to match the gauge. I don't know if that is possible to achieve 1000 TPM. Guessworks will know.