That style of safety (on the cocking piece) was introduced in 1895 for the British service Lee Metford, and remained in use on Lee Enfields until the introduction of the first SMLE (Dec. 1902). It was also used on Lee Metford and Lee Enfield cavalry carbines. Gunmakers continued to offer it on commercial Lee Speeds for a further twenty years. From 1897, however, customers could choose a sliding wrist safety instead.

It's interesting that 1) there's nothing "new" under the sun; 2) what goes around comes back in fashion someday. What I mean my these muddled cliches is that the thumb safety introduced with the SMLE (and later used on commercial Lee sporters) was not the first time such a safety had appeared on British Lees. The very first Lee Metford (the MkI, not the MkI*) introduced in 1889 had a thumb safety in the same location as what would be re-introduced on the SMLE thirteen years later. For some reason I was never too clear about, the authorities decided to remove this safety. So when the majority of MkI rifles were upgraded to the MkI*, the safety was removed, and it's hard to find a Lee Metford with this safety lever still intact. (The only one I have seen in person is on an early commercial Lee Speed action, which, being civilian, never saw an armourer's upgrade to Mk-anything).

From 1891 to 1895 (the MLM MkI* through the MkII), these rifles had no manual safety, other than the half-cock. However, it is likely (but not verifiable) that commercial sporters continued to have the thumb safety during these years). When the cocking piece safety was intrduced in 1895, gunmakers began using it on sporters, along with (a few years later) the swept forward bolt handle (inherited from the cavalry carbine design).