I'm not sure they'd of gotten the tube at all, unless perhaps to connect to a mainline station for a trip to the country? I still feel it more likely they'd of been driven, either by horse drawn or maybe even car (at least to the station) even in 1906.

I remember some rather old photos with very well-dressed people waiting for the tube... so probably "it depends". Compared to horse-drawn carriages, the tube was fast and not prone to traffic jams.