What's the UX in modern airliner avionics for diversion choices, and does it integrate to autopilot?
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.




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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...




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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. 



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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.