Response time depends on where you are and what else is running, but drainage emergencies are triaged: sewage inside a property or a flooding manhole is attended before a slow-draining sink.
When you call, having a few details ready speeds things up: which fixtures are affected, whether any chamber is full, whether the property is on a shared drain, and whether there is vehicle access to the rear.
Every engineer arrives with rods, a jetter and a camera, so most jobs are diagnosed and completed on the first visit.