Just implemented Philip's solution. 

But I'd like to hear someone from Atlassian share how they solved this problem internally for their support desk... as I get what look like automated alerts before a ticket is due to expire, but with what looks like some configurable fields, such as agent name and customer name.

Tieing the automation rule to an SLA 'works' but feels like a major kludge.

This was implemented for Cloud customers back in February 2018 - why hasn't this change been incorporated into the Server edition of Jira Service Desk?

This feature has been implemented in Cloud JSDCLOUD-1861  but it still in "Gathering Interest" for Server ? This feature is a top requested automation and unfortunately we need third party add-ons;

+1 on this one. I can live with Philips solution above, if I had the chance to choose what clocks should be visible on the view screen. Can be quite a few clocks to look at eventually.

We've implemented this using automation rules and SLAs.

We have three SLAs:

Follow up - 4 days
Follow up - 14 days
Resolve - no response

Each of these start counting time when entered status is waiting for customer, and stop counting time when "Comment: by customer".

The automation then fires when each of those SLAs breaches. For the first two, a comment is added and, for the last one, the issue is transitioned to resolved.

It is ironic that this issue is still not resolved, even though Atlassian had this functionality on its very own Support Desk with the message _"We have noticed that this issue has not been updated for a period of 10 business days.
We assume that this issue is resolved. However if this is not the case, feel free to add a comment and we will respond accordingly.
If you've managed to resolve this issue locally would you kindly spare a moment to share your solution for future reference.
You can also request a delay (freeze) now if you need more time to respond."_

Any idea this is implemented?

I agree that this is a "must have" functionality to make the work efficient for many many help desks using Service Desk.
Both the ability to notify users when not updating the issue during a period, and the abilty to transition issues to resolved when getting no updates from customer is needed.

This is the one part of the new automation that is missing for us too. The JIRA Automation plugin allows you to define a JQL query and run it on a cron schedule. The think the only thing missing for Service Desk is a 'Cron Trigger' where you could setup a time to run a automation rule - you could then define the if for timing and the action to comment, close, etc.

Two pre-defined rules - comment that there is no response after x days, and one for auto-close after x days would be great - that has always been one of the major missing pieces to Jira Service Desk versus other service desk solutions.