What's the UX in modern airliner avionics for diversion choices, and does it integrate to autopilot? This question mentions a flight that had to divert, in that case to Iqaluit (YFB).

I was wondering: in modern large airliners, what is the user interface like when diverting? How is the case of "what is the nearest airport/runway we can divert to now?" handled in the avionics displays?

Does the computer system know at all times what the nearest runway or airport is? Does that information appear when needed, or perhaps it's just continually displayed and updated at all times?

Does the system understand subtleties such as which one is best based on your current heading, altitude and so on? Or do pilots just have to quickly look up the nearest one with no special help from the avionics?

Or, is it figured out for each stage, as part of the preflight planning? Do they have it written down as part of a flight plan? Or does the crew have to program in a list of suitable diversion airports along the coming route, and if the worst happens the system will tell you which one (of that set) is nearest now?

In fact, is there just a big red button marked "Divert Now", and the plane immediately changes course to whatever it decides is the nearest and/or best diversion prospect at that moment?

To be clear, I'm talking about how it works in this sort of cockpit:



(To a civilian, most modern airplane controls look like that: touchscreens with some sort of graphical user interface.)
There are a few options for this and it depends on the type of plane you are in, for the most part it looks like the information must at least be proactively navigated to in what ever NAV system you are using.  

Every pilot should study their route before hand and identify possible diversions or at least understand where they are. For pilots flying common trans ocean routes there are some pre-arranged diversion airports that the pilot should know.  

If you are using some kind of tablet as your source of navigation information (maps) most of the modern apps have a function for this. 

If you have some kind of in panel GPS like a Garmin GNS system there is also a page for that very information. 

According to this doc FMS systems may have this info as well. For example in the 737


  This extremely useful page takes a couple of minutes to calculate but
  will list the nearest airports in the database in order of DTG.
  
  Once again, line selection of 1R to 5R will give more useful diversion
  information as shown below.




(source)

You can also program in you alternates 


  You can enter up to 5 alternates here, selecting 1R to 5R against any
  entered alternate will show the info below...




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The FMS will compute data to those points from the aircrafts current location. 


  All the diversion data is now shown based on you flying direct to this
  alternate from present position (VIA DIRECT). Selecting MISSED APP
  will show the same data but calculated from the missed approach point.
  Selecting nearest airports will give...




(source)

Even the more modern airbus cockpits contain a traditional FMS input or possibly a keyboard and mouse. To my knowledge airlines have not really adopted touch screens yet. 



(source)

Some of the smaller panel avionics are starting to incorporate touch screens like Garmins GTN series however some pilots prefer the tactile feel of hard buttons.