depending on the way you turn the question i would say Logitech worldwide although their mechs are maybe not the most popular, as a brand that makes mech i think they may be, them or Cherry, as both are used in offices everywhere mouse and keyboards.
if you were talking only for the mech part of the business i have no clues whatsoever, it very much depend on your definitions of popular and mechanical, in numbers of unit sold IBM/Lexmark/Unicomp may be quite high, in modern market shares maybe a razer or corsair