Mr. Twemlow is a good example of a man who means well but is too terrified to go against the general opinion of British society in any way. His biggest concern in life is whether he is the oldest friend of Mr. and Mrs. Veneering. In other words, he gets through life trying to ride other people's social coattails. As the narrator tells us,
The abyss to which he could find no bottom, and from which started forth the engrossing and ever-swelling difficulty of his life, was the insoluble question whether he was Veneering's oldest friend, or newest friend. (1.2.4)
This isn't a great question to base your whole life on, but Twemlow is a nice enough guy. It'd be nice to see this dude grow a spine before the book ends.
And lucky for us, he totally does. At the end of the book, it's Mr. Twemlow who sticks up for Eugene Wrayburn's decision to marry the working-class Lizzie Hexam. Twemlow gets challenged by the bully Mr. Podsnap, who asks whether Twemlow's rich relative would approve of what he's saying. Twemlow is finally sick of being pushed around by snooty people, so he answers,
Mr. Podsnap […] permit me. He might be, or he might not be. I cannot say. But I could not allow him to dictate to me on a point of great delicacy, on which I feel very strongly. (9.17.54)
It's possible that Mr. Twemlow has bought himself a one-way ticket straight out of London's highest social circles. But he can sleep easy at night knowing that he has stood up for what he believes in. Dickens no doubt hopes that many of his readers will undergo the same transformation as Twemlow.