My countdown code isn't working?

I noticed two problems, I will put a comment beside each one.

local maps = game.ServerStorage:FindFirstChild("Maps"):GetChildren()

local m = Instance.new("Message", workspace)

while wait(1) do
    m.Text = "Starting a new round!"
    wait(3)

    m.Text = "Choosing a random map..."
    wait(3)

    local chosenmap = maps[math.random(1, #maps)]:Clone() --if you don't clone it the map will be gone when you destroy it
    chosenmap.Parent = workspace

    m.Text = "The map is " .. chosenmap.Name .. " by " .. chosenmap.Creator.Value .. "!"
    wait(3)

    m.Text = "Teleporting players..."
    wait(3)

    for i, v in pairs(game.Players:GetPlayers()) do
        if v.Character.Humanoid.Health > 0 and v ~= nil then
            local spawns = chosenmap:FindFirstChild("Spawns"):GetChildren()
            v.Character:MoveTo(spawns[i].Position)
        end
    end

    local roundtime = 60

    while roundtime > 0 do
        m.Text = "Time until next round " .. roundtime
        roundtime = roundtime - 1
    wait(1) --you weren't waiting at all so it just skipped right through this
    end

    m.Text = "The time is up!"
    wait(3)

    for i, v in pairs(game.Players:GetPlayers()) do
        if v.Character.Humanoid.Health > 0 and v ~= nil then
            v:LoadCharacter()
        end
    end
end

In line 30 you start the while loop but you don't have a wait(1) in there so basically it will instantly make roundtime = 0 because there is not wait in the loop

You did not add a "wait(1)" into the loop. Do this:

while roundtime > 0 do
        m.Text = "Time until next round " .. roundtime
        roundtime = roundtime - 1
wait(1)
    end