In the code you provided, the first line is:
var currentdate = new Date();
The constructor of the Date object is called without any parameters, and thus is initialized with a timestamp having the same value as Date.now(). That value is a Unix timestamp, containing milliseconds since 1970-01-01 00:00:00.000 UTC (not accounting for leap seconds). It is strictly UTC based. The sytem time zone is not involved in the construction, and the Date object does not store the system time zone.
Only the system clock is involved. Every computer system has an internal clock that can return the current UTC time. This clock is most often synchronized with clocks of other computers via NTP, PPTP, or similar mechanisms. It is computer dependent though.
For example, when configured for "set time automatically", Windows computers will obtain their time either from a public NTP server on the Internet, or from a domain controller on a local network, depending on the network configuration. However, a standalone computer that is not configured to set time automatically, could have a very different clock value.
Thus, two given computers calling new Date() (or Date.now()) will likely retrieve similar values if their clocks are synchronized correctly, but are not guaranteed to. There is no network call involved, only a call to the local system clock.
In the rest of your code, you call the functions getMonth, getFullYear, getHours, getMinutes, and getSeconds. Each of these function calls will independently use the Date object's UTC-based timestamp and the time zone configured on the computer where the code is being executed. If the code is executed in a web browser, then it's the user's computer's time zone that is applied. If the code is executed on the server, then it's the server's time zone that is applied.
Again, the important thing to remember is that Date object only track a single numeric value - the milliseconds since the Unix epoch (not accounting for leap seconds). You can see this value directly by calling d.getTime() or d.valueOf() on the Date object (named d here), or by coercing the value to a Number as in +d. The time zone is not part of the object's state, and it does not make network calls on its own.