How are landing aircraft handled during an emergency runway closure?
As a tower controller, in case of an unexpected unusual situation for an aircraft on final approach, this will be my list of priorities:


Clear the runway of other traffic
Determine if fire and rescue services need to get involved
(Call fire and rescue services)
Call approach and ask them to stop inbound traffic. In case of other traffic on final behind the emergency, coordinate missed approaches
Instruct other aircraft on final to go around, in accordance with the coordination with approach


The above points will take me no more than 30 seconds to get through. This takes care of the most immediate priorities. I will then await the landing of the emergency plane, and coordinate with fire and rescue as appropriate.

As for departing traffic on the ground - aircraft on their gate will be instructed to stay there. Aircraft taxiing to the runway will either continue to the runway holding point or be turned around to a gate, if available - this depends on whether I expect the runway to be closed for a long time (say, more than 20 minutes). The same, more or less, applies to VFR traffic in the control zone intending to land. They can hold in the air if I expect to reopen the runway shortly, otherwise they might have to find an alternate; I will coordinate any diversions with approach.

Once I have a hold of my own traffic, I will call the operational supervisor and inform them of the situation. They will take care of stopping additional inbound traffic if required, and coordinating with adjacent units and sectors. I might also call approach again directly to let them know how the situation is developing.

It will be up to approach, in cooperation with the supervisor and based on the information I provide, to determine if aircraft on initial approach should be diverted, or if they can hold in the air awaiting a runway reopening. Each individual aircraft will require individual handling, to determine if, where and how they want to divert.

Once the runway is reopened (after fire and rescue have finished their work, the emergency airplane has been removed, runway inspection carried out and so on), I will coordinate with approach and the supervisor to start inbound traffic again (possibly with a reduced rate), and start getting my own departures rolling again.

As you can see, there are a lot of "if's" in the above. A lot of decisions have to be made, and the basis for making said decisions changes depending on the situation. There is no set checklist or rulebook for when something like this happens, because the required actions will vary. But that is what we get paid for after all - as an air traffic controller you have to be creative and flexible, especially when unusual situations occur.