I guess there are two things it would need.
1/ Firstly it would require the teacher or admin, or whoever had to authority to nominate the next class that waitlisted students move into.
I guess if it was a class that repeated, it could have a list of the next set of classes with radio buttons for example, so the student has a choice of which class they would like to attend. Just in case the next class was at a time unsuitable, yet another class time was suitable.
2/ As far as payment goes, I would presume it would again need an option set by the administrator, depending on their system. Maybe the student has a choice. They can have two options for example. Paypal or manual payment collection.
If Paypal, it takes them to a screen that allows them to pay and be secured into the class. (I presume this plug-in would work with a Paypal plug-in, unless it is better to add paypal to this plug-in)
If manual, they are asked to confirm their user details, especially their email address and phone number, and a confirm button sends them an email saying they have asked to pay for a course and an admin person will contact them shortly for payment, and it also sends admin(or who ever is responsible) an email advising the student would like someone to call to take payment. The admin takes the payment over the phone and manually confirms the enrolment in that class.
To save confusion, the automatically generated email that goes to admin would also have to show the course the student selected from the list of available courses. When the admin confirms the enrolment, an email is sent to the student advising them of the course enrolment and the dates the course will be available etc.
Does this sound like a reasonable scenario?