Service in the SAMIF was a way to get into the Anglo Boer War at great speed, it could be very rewarding financially too, many men came from across the empire and in some cases, actually well beyond it, so nothing unusual here whatsoever, of course, he could have merely worked his passage, but, perhaps the first thing to do would be to have a careful look through the passenger lists in the Board of Trade series at Kew, almost all of it is available on line these days, but, however the method used by Brady, the end result is the same.
He joined the ranks of the 1st Kitchener's Fighting Scouts at Durban on the 6th of April 1901 and was discharged there as a Farrier Corporal on the 10th of October that year, WO100/256 and 257 confirm his medal and four clasps were issued on the 23rd of December 1903.
Surviving rank and file papers for this particular regiment are in WO126/76-82, so it would be worth taking a look for those at some stage.
The 1st KFS was a particularly good regiment, five squadrons in strength, raised in Pretoria on the 23rd of November 1900, standard enlistment was for six months, they were to see some really intense fighting in the Northern Transvaal and were witnesses to the wars most brutal phase out on the veldt, being disbanded in Johannesburg on the 7th of July 1902.
Men like Brady brought about the peace that came in 1902, the hard way, you can forget about Talana and such places, the enemy were to be ultimately bled dry in the Transvaal.