What is a normal wait time to get an appointment for emergency work? For a non-emergency?

Each plumbing business will have different workloads and existing customers. The expected wait time could depend on this, in addition to the nature of the emergency. If you have a major leak, turn your water off at the internal stop tap (normally under your kitchen sink) and open all taps to drain the system. Even in an emergency, you may want to contact more than one plumber to get an idea of how soon they can schedule a visit.

For non-emergency work you should agree a suitable time for both yourself and the plumber for an appointment, bearing in mind this could be an evening or weekend.