How come the value wont get changed and I wont get warned because of it?

Made a new variable since in line 36, it specified "PlayerCountries" and in the last code (your code), it was specified the cloned instance to be the parent of "PlayerCountrys" at line 30 (Just fixing a bit of spelling in mine)

I see no errors other than that. It should work just fine even in your script; it just means the script couldn't find the instance and then skips that line of code, then the next executes.

local Player = game.Players.LocalPlayer
local ReplicatedStorage = game:GetService("ReplicatedStorage")
local workSpace = game:GetService("Workspace")

local Buttons = script.Parent.Parent.MainGui.Buttons

local ValuesFolder = script.Parent.Parent.ValuesFolder
local PlayerCountries = ReplicatedStorage.PlayerCountries

if Player.Character then
    if PlayerCountries:FindFirstChild(Player.Name.."Country") then
        local PlayerCountry = PlayerCountries:FindFirstChild(Player.Name.."Country")
        if ValuesFolder.Country.Value == "" then
            ValuesFolder.Country.Value = PlayerCountry.Value
            if PlayerCountry.Value == "" then
                warn("Did not work as intended")
            end
        end
    elseif not PlayerCountries:FindFirstChild(Player.Name.."Country") then -- Dunno if you need this one, you can delete this elseif statements if you'd like.
        local PlayerCountry = ValuesFolder.Country:Clone()
        PlayerCountry.Name = Player.Name.."Country"
        PlayerCountry.Parent = PlayerCountries
    end
elseif Player.Character == nil then
    if PlayerCountries:FindFirstChild(Player.Name.."Country") then
        local PlayerCountry = PlayerCountries:FindFirstChild(Player.Name.."Country")
        if ValuesFolder.Country.Value == "" then
            ValuesFolder.Country.Value = PlayerCountry.Value
            if PlayerCountry.Value == "" then
                warn("Did not work as intended")
            end
        end
    elseif not PlayerCountries:FindFirstChild(Player.Name.."Country") then
        local PlayerCountry = ValuesFolder.Country:Clone()
        PlayerCountry.Name = Player.Name.."Country"
        PlayerCountry.Parent = PlayerCountries
    end
end

But if you're trying to execute the script after the instance has been added, then you can wrap it up in a function and then call the function. Or you can wrap it up using coroutine.

DevHub: coroutine

DevForum: What is Coroutine? by Mysterious_Myth11

DevForum: Coroutines - When and how to use them by ReturnedTrue

Youtube: Advanced Roblox Scripting Tutorial #33 - Coroutine by TheDevKing

Bit of advice: you can use "not" in an if statement

local bool = false
local nonExistent = nil
local Frame = script.Parent

if not bool then
    print("false") -- Will print
else
    print("this wont print because bool is false") -- Won't print
end


if nonExistent then
    print("This won't print because it is nil")
else
    print("This will print because it is nil")
end

Frame.Visible = not Frame.Visible
-- This turns the frame visibility boolean to the opposite boolean, e.g: true to false, false to true

The value isn't set to "" bye default its set to nil so try something like this:

 if ValuesFolder.Country.Value == nil then
                ValuesFolder.Country.Value = PlayerCountry.Value
                if PlayerCountry.Value == nil then
                    warn("Did not work as intended")
                end

and if I missed anything else that was set to "" try setting it to nil.