This question is never directly stated in the story itself, so you will have to use inference, or educated guesses, based on clues throughout the story. A good place to start is by looking a bit at the author's life and when he was alive. Lawrence was alive in the end of the 19th century, and the first part of the 20th century. He was also born and raised in England, and most of his novels are set there, at least in part. So, given those details, if you were to guess just based on the life of the author, a good inference to make would be that the story is probably set in the early 1900's in England.
The next thing to do is to look at the story itself to see if the details match what you have decided so far. In this story, the family has a nursery governess, talk about private schools for their children, and use quaint English phrases such as "a bird in the hand is better than two in a bush." The mother and father are constantly trying to "keep up appearances" even though they don't have the money to do so, and his mother hesitates to find work, but occasionally lowers herself into attempting to make money for the family. Given all of these clues, it is safe to assume that the story is indeed set in England, in a higher-class neighborhood where the family is pressured into maintaining a wealthier lifestyle. The lower classes in England did not have such luxuries as nannies or servants, and certainly did not have the time or ability to send their children to fancy schools.
The time period is more open; given that Lawrence wrote about times that were evident in his lifetime, and that nothing in the story contradicts our previous guess of the early 1900's, that is the best that we can do with that as far as the setting goes. I hope that those thoughts help a bit; good luck!