Well, not necessarily instantaneous as we don't know how long it was between when the call was made and when Cobb walked into the house. It could have been anywhere from an hour (assuming the call was made right before landing and Cobb lives near LAX) to the better part of a day (since the time experienced in the Level 1 dream was less than a day that would be less than two hours in reality of a 10 hour flight and then for all we know Cobb and Caine left LAX and drove to the family's home in Yreka).
And Saito may very well have spent the last three months putting in place all the pieces so that they could be knocked over very quickly. Maybe 30 seconds after his call, Saito's agents were in Grandma Mal's living room showing her the proof of how he was innocent, other agents were at the sheriff's office showing the proof to them with a promise that Cobb would be in on Monday to answer any final questions, and a third agent was quickly changing the necessary databases so that Cobb could get in the country without hassle.
I have no idea what was done but, if we accept as given that the last scenes are reality and not dream, then whatever Saito did had to be completed (though not necessarily started) very quickly and it had to be more than just getting him past ICE since returning to America does him no good if he then immediately gets arrested by the local sheriff for murder. Actually getting into the country would never have been a problem, it was the murder warrant that was the problem.
If it was all a dream, then yeah, none of it has to make internal sense. If it wasn't a dream then I don't think it was necessary for Nolan to spell out each of the steps I listed since they are, to me, implicit in what was shown to be happening and nothing shown, as least in relation to this, is a giant plothole (at least to me).
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