MQ00 Stage 5 suggests that some things shouldn't start when the game starts.
; called by MQ102 during chargen - place to put "startup" quests that can't start enabled for various reasonsHousePurchase.Start()
So, I've updated my OnUpdate () function to check for this.
The problem I found with all my mods is that the player gets stuck in the auto-walk towards the chopping block in the prologue scene. Something I was doing in this procedure was causing the problem. Adding "If questMq00.GetStage () >= 5" in my OnUpdate () block fixed it.
Note that the problem doesn't present itself after you've chosen to follow either Ralof or Hadvar.
Note to new users, the methods suggested in this thread are better by Justiinother for GetGameLoaded () and by RandomNoob for GetGameRestarted (): http://www.gamesas.com/topic/1345542-getting-a-script-to-run-when-the-game-loads. To summarise: the checks respectively uses (1) changing an Actor Value that is never saved and (2) adding a Scripted Spell to the player. However, I've not implemented or tested neither and have stuck with my method (which checks for changes in Game.GetCurrentRealTime ()).
I'm sure that future versions of SKSE will have non-hacked versions of OnGameLoaded () and OnGameRestarted ().