Peter and Wendy, also called Peter Pan, is the title of a play published in 1904 as well as a novel adaptation published in 1911. Both the play and novel format tell the story of Wendy Darling, her brothers Michael and John, and a mischievous, magical boy called Peter Pan. Wendy and her brothers are invited to live in Neverland with Peter and his gang of Lost Boys. Even though they are impressed with Neverland and its fascinating inhabitants, the Darling children decide that they would much rather live at home with their parents and dog.
There are a few things to take into consideration in determining the time period for Peter and Wendy. First, the play and novel were published in the first decade of the 20th century. It would be easy to assume based on this fact alone that the story of Peter Pan is set around the time of its publication. When we consider some of the context clues, such as the Darling family dynamic as well as pop culture imaginings of pirates and Native Americans, a setting in the late 19th century or very early 20th century seems appropriate.
That the Darling family, who live in London, employ domestic servants (such as a Nanny or Housemaid) would suggest they are of higher economic status. However, we only hear mention of one domestic servant and the children are tended to by either their mother or the dog called Nana. One would assume they are firmly of the middle class based on their ability to employ just the one domestic servant.
Wendy also embodies many of the values emphasized for young women during the Victorian period. Girls and women were encouraged to be maternal, caring, graceful, polite, tidy, and patient to the point of being passive.
My best guess is that Peter Pan is set in the Edwardian Period, which lasted from 1901 to 1910, though it is possible for it to have been set earlier in the Victorian Period.