As shown in the paintings and photographs linked in her wikipedia biograph: https://en.wikipedia.org/wiki/Elizabeth_II
From my observations she isn't shown with 'red' hair as such, but rather a golden or strawberry blonde. In these images she is very young and as people age their hair tends to darken, particularly if born with hair within the blonde ranges. Know its only a basic explanation but it would probably be better answered by someone in the scientific field.