I am not sure, you still require an answer for this or not. But in your case, this might be the reason:
The videos you're trying to watch is getting viewed in UK as you said you're in UK. Please note, when you're trying to watch a video or browse anything on the web, your IP Address requests the access to that content. When your request goes to a server having your requested content, it gives the content back to your IP address. Your IP address is your ID on the network through which server knows you and can communicate with you.This is the basic principal of networking.
So, when you're watching a video in UK, you have an IP address that belongs to UK not NZ as you're connected to internet via UK ISP. When your request reaches to the server, the server denies your request because that server doesn't let people in UK to access its information. (Some websites restrict the access to its content to certain countries).
Please note this has nothing to do with your Computer's account profile (like time and date as you said). This gets stored in CMOS (a hardware that's present inside your computer) and is local to your laptop. So, you can freely change it to any country or time zone.
I hope this makes sense if you still needed a help.