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.
relative to their wealth
But total wealth was a lot lower than today, while clothes were much more expensive relative to the average income than today. Many people wore their best clothes on Sundays only, and street photos from the same period show that an average person was dressed rather simply.
I don't even think that the tube was affordable to average working class public. When I read historic accounts from much later (the 1930s), they mention that fare on public transport was expensive enough that you had to choose between a breakfast and a ride.
Other way round is my understanding. Road based transport was for the poor, especially trams, and the underground was for the fancy folk looking to shortcut past all of that.