WebBrowser.getCurrentDate() should work (based on an measurement/estimation of the time difference between server and client, returning the time in the server’s time zone).
However, you can also check WebBrowser.getTimezoneOffset() and WebBrowser.getRawTimezoneOffset(). You can then get GMT time from the server and apply the adjustment getTimeZoneOffset() assuming the client computer is correctly configured.
What if the user changes their local time and they are on a completely different date? For example, for testing. Can we get the actual time, not assuming that it is in sync with the server time?