Pulling the countdown logic out of the GameManager and the LocalLobby; the latter fixes a bug where the countdown would prevent any lobby changes from being pushed from the host, since every from the LocalLobby's OnChanged event would fire.

Assets/Prefabs/UI/PlayerInteractionPanel.prefab

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   m_Name: CountDownUI
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Assets/Scenes/mainScene.unity

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Assets/Scripts/Game/GameManager.cs

             {   m_localUser.UserStatus = (UserStatus)msg;
             }
             else if (type == MessageType.StartCountdown)
-            {   BeginCountDown();
+            {   m_localLobby.State = LobbyState.CountDown;
-                m_localLobby.CountDownTime = 0;
+            }
+            else if (type == MessageType.CompleteCountdown)
+            {   if (m_relayClient is RelayUtpHost)
+                    (m_relayClient as RelayUtpHost).SendInGameState();
             }
             else if (type == MessageType.ConfirmInGameState)
             {   m_localUser.UserStatus = UserStatus.InGame;
             {   m_localLobby.State = LobbyState.Lobby;
-                m_localLobby.CountDownTime = 0;
                 SetUserLobbyState();
             }
             else if (type == MessageType.QuickJoin)
             else if (type == MessageType.SetPlayerSound)
             {
                 var playerSound = (LobbyUserAudio)msg;
-        }
+            }
         }
 
         private void SetGameState(GameState state)
             OnReceiveMessage(MessageType.LobbyUserStatus, UserStatus.Lobby);
         }
 
-        private void BeginCountDown()
-        {
-            if (m_localLobby.State == LobbyState.CountDown)
-                return;
-            m_localLobby.CountDownTime = 4;
-            m_localLobby.State = LobbyState.CountDown;
-            StartCoroutine(CountDown());
-        }
-
-        /// <summary>
-        /// The CountdownUI will pick up on changes to the lobby's countdown timer. This can be interrupted if the lobby leaves the countdown state (via a CancelCountdown message).
-        /// </summary>
-        private IEnumerator CountDown()
-        {
-            while (m_localLobby.CountDownTime > 0)
-            {
-                yield return null;
-                if (m_localLobby.State != LobbyState.CountDown)
-                    yield break;
-                m_localLobby.CountDownTime -= Time.deltaTime;
-            }
-
-            if (m_relayClient is RelayUtpHost)
-                (m_relayClient as RelayUtpHost).SendInGameState();
-        }
-
         private void SetUserLobbyState()
         {
             SetGameState(GameState.Lobby);
         {
             m_localLobby.CopyObserved(new LocalLobby.LobbyData(), new Dictionary<string, LobbyUser>());
             m_localLobby.AddPlayer(m_localUser); // As before, the local player will need to be plugged into UI before the lobby join actually happens.
-            m_localLobby.CountDownTime = 0;
             m_localLobby.RelayServer = null;
         }
 

Assets/Scripts/Game/LocalLobby.cs

             get { return new LobbyData(m_data); }
         }
 
-        float m_CountDownTime;
-
-        public float CountDownTime
-        {
-            get { return m_CountDownTime; }
-            set
-            {
-                m_CountDownTime = value;
-                OnChanged(this);
-            }
-        }
-
         ServerAddress m_relayServer;
 
         /// <summary>Used only for visual output of the Relay connection info. The obfuscated Relay server IP is obtained during allocation in the RelayUtpSetup.</summary>

Assets/Scripts/Infrastructure/Messenger.cs

         QuickJoin = 14,
         ClientUserApproved = 15,
         ClientUserSeekingDisapproval = 16,
+        CompleteCountdown = 17,
     }
 
     /// <summary>

Assets/Scripts/Infrastructure/Observed.cs

             onChanged?.Invoke(observed);
         }
 
-        protected void OnDestroy(T observed)
+        protected void OnDestroyed(T observed)
         {
             onDestroyed?.Invoke(observed);
         }

Assets/Scripts/UI/CountdownUI.cs

     /// This countdown is purely visual, to give clients a moment if they need to un-ready before entering the game; 
     /// clients will actually wait for a message from the host confirming that they are in the game, instead of assuming the game is ready to go when the countdown ends.
     /// </summary>
-    public class CountdownUI : LocalLobbyObserver
+    public class CountdownUI : ObserverBehaviour<Countdown.Data>
-        protected override void UpdateObserver(LocalLobby obs)
+        protected override void UpdateObserver(Countdown.Data data)
-            base.UpdateObserver(obs);
-            if (observed.CountDownTime <= 0)
+            base.UpdateObserver(data);
+            if (observed.TimeLeft <= 0)
-                m_CountDownText.SetText($"Starting in: {observed.CountDownTime:0}");
+                m_CountDownText.SetText($"Starting in: {observed.TimeLeft:0}"); // Note that the ":0" formatting rounds, not truncates.
         }
     }
 }

Assets/Scripts/Game/Countdown.cs

+using System;
+using UnityEngine;
+
+namespace LobbyRelaySample
+{
+    /// <summary>
+    /// Runs the countdown to the in-game state. While the start of the countdown is synced via Relay, the countdown itself is handled locally,
+    /// since precise timing isn't necessary.
+    /// </summary>
+    [RequireComponent(typeof(UI.CountdownUI))]
+    public class Countdown : MonoBehaviour, IReceiveMessages
+    {
+        public class Data : Observed<Countdown.Data>
+        {
+            private float m_timeLeft;
+            public float TimeLeft 
+            {
+                get => m_timeLeft;
+                set
+                {   m_timeLeft = value;
+                    OnChanged(this);
+                }
+            }
+            public override void CopyObserved(Data oldObserved) { /*No-op, since this is unnecessary.*/ }
+        }
+
+        private Data m_data = new Data();
+        private UI.CountdownUI m_ui;
+        private const int k_countdownTime = 4;
+
+        public void OnEnable()
+        {
+            if (m_ui == null)
+                m_ui = GetComponent<UI.CountdownUI>();
+            m_data.TimeLeft = -1;
+            Locator.Get.Messenger.Subscribe(this);
+            m_ui.BeginObserving(m_data);
+        }
+        public void OnDisable()
+        {
+            Locator.Get.Messenger.Unsubscribe(this);
+            m_ui.EndObserving();
+        }
+
+        public void OnReceiveMessage(MessageType type, object msg)
+        {
+            if (type == MessageType.StartCountdown)
+            {
+                m_data.TimeLeft = k_countdownTime;
+            }
+            else if (type == MessageType.CancelCountdown)
+            {
+                m_data.TimeLeft = -1;
+            }
+        }
+
+        public void Update()
+        {
+            if (m_data.TimeLeft < 0)
+                return;
+            m_data.TimeLeft -= Time.deltaTime;
+            if (m_data.TimeLeft < 0)
+                Locator.Get.Messenger.OnReceiveMessage(MessageType.CompleteCountdown, null);
+        }
+    }
+}

Assets/Scripts/Game/Countdown.cs.meta

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