I want to show the timezone code such as BST or GMT based on the user's locale. However in each browser it is positioned in different places from new Date().toLocaleString() and on some browsers (such as Opera) not available at all.
It looks like the solution would be to get a database or structure of timezone codes against timezone offset but there can be multiple matches (e.g. BST == GMT) and I can't even find such a list.
Does this mean it can't be done?
You have to manually define the timezone differences in your code. It is complicated because GMT time is different than UTC (Zulu) time. GMT includes a change for daylight savings time, while UTC has no daylight savings and is otherwise the same timezone representation.
Additional confusion is that not all political zones define their timezones to vary by an example number of hours. India is UTC + 5.5 hours and Afghanistan UTC + 4.5 hours. Notice that is UTC and not GMT as the base as those countries do not observe daylight savings. Some small island nations in the southwest Pacific can vary on the quarter hour.
Then some political timezone definitions have internal differentiation. Arizona is typically considered to be in the US Pacific timezone, GMT - 8 hours, except that it does not observe daylight savings, so half the year it is in US Mountain timezone as Arizona is really UTC - 8.
With all that confusion there are hundreds of definitions for timezones, and some of those change as political boundaries change. I just say screw it and use either the user's clock time or adjust to UTC time by dynamically adjusting a constant in your JavaScript from the server to reflect a known value that represents UTC time when the user requested the page.