This is a great question, as I personally think when we can compare two novels that are from such different locations, cultures and time periods, it helps us to see the universal over-arching themes within nature that make it such an essential part of human existence.
The biggest similarity I would look at in reference to these two novels is the way in which both show how attitudes between generations have changed and developed. In this novel by Dickens, Thomas Gradgrind is forced to concede that he actually did too good a job in bringing up his children to focus on facts alone to the exclusion of all else. He has to confront the fact that largely thanks to his parenting, Louisa is emotionally inaccessible and unable to reach out to others, and that his son, Tom, referred to affectionately as "the whelp," is a failure in life.
In the same way, in When I Whistle, Ozu has to come to terms with the radically different outlook and perspective of his son Eiichi, who is an unscrupulous doctor. In this novel, the difference between the generations as a result of the war is focused on, and how it was accepted that the younger generation had to adopt much more unsympathetic measures in order to get ahead. Note what Ozu's wife says of her son's lack of empathy with his patients and the way that he uses them:
Times are different... Young people now can't survive if they don't push others out of their way. There's really nothing else Eiichi can do.
Ozu thus has to confront the way that his son uses people to get ahead, even going as far as giving untested drugs to patients who are very ill indeed.
Both novels then focus on the struggles of one generation to understand their children and the very different kinds of lives that they experience compared to their parents.