As soon as you are registered for the prerequisite course at the University of Arizona for Winter or Summer, you should be able to register for the subsequent course for the following semester.
Note: The timing is important!
Summer registration usually opens at the beginning of Priority Registration for Fall (after Spring Break). By the time you are able to register in your Priority Registration week, you should be able to register for the subsequent course in Fall with no problem.
Winter registration usually opens AFTER Priority registration for Spring, i.e. when Open Registration for Spring begins. This means that you will need to wait until open registration to 1) register for the Winter prerequisite class, then 2) register for the Spring class.