We are first introduced to Tellson´s Bank in Chapter 1 of the second section of the novel, "The Golden Thread". The description we are given of Tellson´s Bank is worthy of some study, and you will find the answer to your question there.
It is described as "old-fashioned" twice in the first paragraph, and then the second paragraph gives us a key indicator as to their attitude towards change:
Any one of these partners would have disinherited his son on the question of rebuilding Tellson´s. In this respect the House was much on a par with the Country; which did very often disinherit its sons for suggesting improvements in laws and customs that had long been highly objectionable, but were only the more respectable.
Thus Tellson´s Bank is established as an institution that clings to the past and will not "improve" in any way. The description we are given of its interior is funny in the uncomfortableness of its environs. Upon entering Tellson´s Bank, we are told, you come to your senses in a "miserable little shop", with the "dingiest of windows" and customers who need to see "the House" they are made to wait in the most "dismal twilight" contemplating their lives as if it were a prison. The general description is of squalor and old age, for we are told:
When they took a young man into Tellson´s London house, they hid him somewhere till he was old. They kept him in a dark place, like a cheese, until he had the full Tellson flavour and blue-mould upon him.
Whilst this description of the "maturing process" of Tellson´s workers is funny, it contributes to the overall picture of an extremely traditional institution that is definitely not open to new ideas or change.