It depends entirely on the particular President and policy, you can't really generalize this sort of thing.
It depends from the kind of policy. The effects of Clinton's budget policies were already visible by 1996, and the effect of the Irak War had become obvious by 2006 or so.
I know this.
Obviously some policies are immediate in their application: the Emancipation Proclamation freed all slaves on American soil immediately, and I do not question this immediacy.
The problem I have, however, is that there are no readily obvious benchmarks to say that "X policy was enacted at such a time and achieved its goal at this other time" with most policy decisions.
In part, this is because the Presidential system is opaque; I think the public is accustomed to thinking that Presidential policies basically end with the term of the President who initiated them, and the changeover in personalities occupying the office serves to obscure the fact that this isn't so. Which is among the reasons I prefer direct democratic systems - personalities wouldn't 'get in the way' of observing policy outcomes over time.
EDIT: Let me explain my problem with this.
We can understand individual Administrations fairly well, and we can weave historically local narratives pretty easily (the 'Reagan Revolution' or the 'New Deal', say). Policy is more-or-less comprehensible on the small scale. But like Hemingway said, the past isn't even past - after so many items in a set, it seems to me to become increasingly difficult to separate one item from another.
The New Deal is obviously partially intact; Social Security still exists, for instance. But were Social Security's solvency issues 'baked into' the programme when it was first created, or are they the result of tinkering over decades and Administrations of both Parties? It's difficult to impossible to say, and the evidence is solid for both sides. This ambiguity is the result of the opacity I mentioned earlier.