Is the whole of the UK in the same time zone? Is that the "0th" time zone?
TIA
- Appears so. 2. Yes.
Image source: https://geography.name/time-zones/
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What does this have to do with Arduino?
On the subject of the time zone of the UK, you should be aware of the existence of British Summer Time.
Yes, we are.
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Well, I'm working on a 1 line LCD clock that lets me select a country or region with UP / DN buttons and display that local time of day for 5 seconds, then return to local time. I had a Pro Mini in mind but if you feel an Arduino is inappropriate, I can go with a Teensy or...
BTW, we have "Summer time" as well, AKA Daylight Savings Time.
The sun never set on the British flag...
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Thank you (and Perry) for a simple, direct answer to a simple, direct question and wish you a great New Year.
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do you count the British Overseas Territory into the UK?
(for example not to go very far away Gibraltar changed to use Central European Time all year round in 1982)
Wouldn't be easier to look it up on Wikipedia?
The 'UK' is 'The United Kingdom of Great Britain and Northern Ireland', it does not include British overseas territories.
note to self: (UK != British Empire)
Right so only England, Scotland, Wales and Northern Ireland.
I always get confused 
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Note, that not all countries that have Daylight Saving Time make the change from summer to winter on the same date - a further complication for your project.
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Noted ,TNX
Have you tried?
I assume local time for that zone, for example I want to respond to a poster in NZ but not at 0300 there, I could press button until "NZ" was on display, stop for 0.5 seconds, then NZ time would show for 5 seconds, then revert to local time.
I now tried with chatGPT with your question
Q: Is the whole of the UK in the same time zone? Is that the "0th" time zone?
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A: The United Kingdom is not entirely in the same time zone. The standard time zone for most of the UK is Greenwich Mean Time (GMT), which is often referred to as the "0th" time zone since it is located along the Prime Meridian at 0 degrees longitude. However, during daylight saving time (usually from the last Sunday in March to the last Sunday in October), the UK observes British Summer Time (BST), which is GMT+1.
So, while the majority of the UK follows GMT or BST, there are some variations. For example, the Shetland Islands, part of Scotland, may experience slightly different local times due to their geographical location. Overall, GMT and BST are the commonly used time standards in the UK.
upon further asking for the Shetland Islands, it confirmed there is nothing special
TNX for that, I'm getting more education than I expected. 
Because God never trusted the British in the dark.
ChatGPT reminds me of a quote about Gilbert & Sullivan operas "full of words and music and signifying nothing"
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seems about right
OK, now I understand what you're trying to do.
What you are trying to do is really an easy, trivial task, but for one thing: Daylight Saving Time.
You will have a lot of research to do to find out what regions use Daylight Saving Time, and when they use it. You will then have to adjust for Daylight Saving Time when, and only when, it is in effect in those regions. I also suggest that you allow your clock project to be re-programmable in case the rules change (which they probably will, somewhere, at some point).
If you use a 2-line display, you can show your own local time on one line and the selected time zone on the other line. But, it's your project, so if you want just 1 line, then 1 line it is.
I strongly suggest you start simple. First, forget about Daylight Saving Time. Can you build a clock that just shows UTC? (You know what UTC is, right? You might also know it as "Zulu time" or "GMT".) If you cannot build a clock that just shows UTC, then you have no hope of taking this project any further.
For this project, you probably need a DS3231 (google it if you don't know what it is). It is a chip that keeps accurate time, because an Arduino alone is not meant to be an accurate timekeeper. You will need to learn how to set the time on a DS3231, and read the time from a DS3231.
You will probably want to have your DS3231 set to UTC. Once you accomplish that (there are several methods for setting the time on a DS3231), just try getting that time to appear on your LCD display. Then add 8 hours to UTC and you have the time in China. (China does not use Daylight Saving Time.) Add 9 hours to UTC to get the time in Japan and in South Korea. (Japan does not use DST either. Neither does South Korea.) Learn about buttons and how to debounce them. Then you can use buttons to switch between UTC and China time and Japan / South Korea time.
To add more time zones, you will need to do Daylight Saving Time math.
Here is a post I wrote in which I go into that a bit (DST for the USA):
Here is another post I wrote (DST for Greece):
You will need to do different math for different countries / time zones. I might be able to help you with some of the math, because of how my brain works. But this is still your project, so you will be the one to put it all together.
TL;DR: This will be a big project. You can do it, but it will be a lot of work.