On Windows 10 / Server 2012 / 2016 I can set the "short time format" by using the Get-UICulture and Set-Culture commandlets to the 12 hour format with AM/PM indicators.
However, I can not change it from 12 hour format to 24 hour format. It remains in the 12 hour format - meanwhile there is no problem to set the 24h format from the Settings app in Windows.

(In contrast to that, the LongTimePattern works fine both ways)

Steps to reproduce

(Get-UICulture).DateTimeFormat | Select ShortTimePattern,LongTimePattern | fl
$modCulture = Get-UICulture
($modCulture.DateTimeFormat).ShortTimePattern = "h:mm"
($modCulture.DateTimeFormat).LongTimePattern = "h:mm:ss"
Set-Culture $modCulture

(Get-UICulture).DateTimeFormat | Select ShortTimePattern,LongTimePattern | fl
($modCulture.DateTimeFormat).ShortTimePattern = "HH:mm "
($modCulture.DateTimeFormat).LongTimePattern = "HH:mm:ss"
Set-Culture $modCulture
(Get-UICulture).DateTimeFormat | Select ShortTimePattern,LongTimePattern | fl

Expected behavior

It should change the short time format to 24h format "HH:mm"

Actual behavior

The time format never gets set to 24h format. It remains in the AM/PM 12 hour format.