Greg:
I did not have this issue with the above code saved as a FreeCDF file using CDFPreview. At each opening of the CDF file the data was updated.
The following is tangential to your immediate issue you wish solved.
DynamicModule[{t = None},
Column[{Dynamic[
ActionMenu[
"Update Time:", {"On" :> (t = (UpdateInterval -> 1);),
"Off" :> (t = None;)}, Method -> "Queued"]],
Dynamic[Refresh[
Row[{ClockGauge[AbsoluteTime[TimeZone -> 0],
PlotLabel -> Style["Student", Large, Bold]],
ClockGauge[
AbsoluteTime[
DateList[
Lookup[Association[
Values[URLRead[
HTTPRequest["http://www.google.com"]][{"Headers"}]]],
"date"]]], PlotLabel -> Style["Server", Large, Bold]]}],
t]]}, Alignment -> Center]]

Now this worked each time I open the cdf file. It is a bit slow upon opening the first time. But it updates. The external URL is different from the above. But HTTPRequest seems to work in a CDF file. I am not using CDF Player. Other ideas are DockedCells, Using CloudDeploy or APIFunction to deploy an API call that returns a current timestamp. There are examples in the documentation.