This is part one of two related questions.
Let us suppose that we have an expert in a a particular period of time, a person who is one of the world's foremost experts on the period. This person also has the ability to speak the local language and blend in with the locals.
This person travels back in time to the period in question and does... nothing. They simply live as a local and observe events as they occur. This means that they must interact with the locals, may even form relationships with some, but do not reveal their origin, and do not attempt to change the course of events.
In theory, as soon as the time traveller appears in the past timeline, the effects of their presence will ripple out at the speed of light, affecting events at the quantum level, altering the outcome of random events. However, humans are macroscopic structures, and not quite as easily influenced. However, the vagaries of random chance will accumulate, and eventually have a noticeable effect.
How long is it likely to be before our expert notices divergences from their memory of the historical course of events, and how long before the world and its history is no longer recognisable?
While this question is applicable to time travel to any place or time, it is easier to have detailed knowledge of more recent times.
This is considered to be a "many worlds" scenario - the time traveller cannot invalidate their own existence or presence. They are also not trying not to have an effect on the course of major events, they are just not trying to have an effect.
We can presume that the observer is observing a major world event such as a war or diplomatic negotiations at close range. They are not "out of the way", just not getting involved.
As to the effect a time traveller might have, their mere presence will have all sorts of effects. At the very least, their presence may lead to the weather being different over the course of some weeks or months. Personal interactions can lead to all sorts of changes. Maybe the time traveller bumps into someone and stains their clothes, and as a result that person is not taken as seriously by their superiors, or a rider guides his horse around the time traveller, and as a result the horse doesn't step on a stray nail and become lamed...
While this question should be applicable to any time period, for the sake of example, let us assume that the time traveller appears in Europe at the outbreak of WWI.
Also, the question is not "what might change?", but "how long would it take before the changes became noticeable and how long before everything was completely different?"