In "Bell of Africa" he says "I acquired two sporting models of the .303, each with the ten shot magazine. In fact they were the military arm with the barrel cut down a bit, and sporting pistol grip stocks. They cost 8 pounds each"
eight pounds was 160 shillings. In my 1913 BSA catalog, 165 shillings would get you a No 3 pattern. ie the basic lee Speed with pistol grip stock.
It would appear that this was his Lee Speed of choice.

In "the wanderings of an elephant hunter" he has one chapter devoted to his rifles and how to hone your shooting skills. a very interesting read. cheers