Battery Saver countdown when at near full battery and I'm working at my desk.

It sometimes happens to me , it is as if somehow one has entered an activity.

As you may be aware , if you open the activity screen  then it starts a lot of processes such as increasing power to OHR , GPS etc etc. hence why there is a power save timeout setting per activity (normal or extended). before it warns one that the activity is open and basically 'shuts down' these additional processes. That why I use extended on some of my activity settings (like race) so it doensn't happen too frequently whilst waiting for a race to start. .

I think that somehow (and I have quite worked out how) the the watch gets into an activity mode and hence after a certain number of seconds this alert is triggered.

I was wondering if my wrist bumps the activity start button , but I've never noticed a specific activity screen being on before this has happened.

You entered an activity profile and you are at the pre start screen. Thats the only way the countdown will appear. 

As you did not start the activity, the watch will go back to watch screen to disable WHR high power mode, disable GPS and disconnect the sensors. 

Wrong. It happens when I haven't touched the watch in an hour, such as when I'm sitting at my desk working.

It’s not wrong. There is no timer outside an activity. If the battery is low, there will be an info screen that will ask if you want to enable battery saver mode. 
Countdown for power safe is only inside an activity. 
And the proof that you have been in an activity (however this happened) is why you are in the activities screen after you push back. 
maybe your start button is broken and is sending signals without pressing. 
wouldn’t be surprised if that would be the root cause. 

I would try:

- forced restart (Hold light button)

- restore watch

- replace watch