A function that will be called after all Start()-s did

I have a seperate scheduler loop that should only start processing after all objects are initialized(started).

Are there any options to do this, except a one-time-call of that function from Update/FixedUpdate(since those are called after all Starts did)?

Start gets called after all Awake’s have been called. So you may want to do it like that. One other way, as you mentioned, is to do a one-time function call from Update, as Update will be run after all Start’s finished.
You could also have a “LastStartCall” script attached to an empty game object and increase its index in the execution order such that it will be executed last. This Start would then be called after all other Start’s.