How does the ground effect affects longitudinal stability?
It depends on general geometry of the aeroplane:


The vertical position of the main wing and the horizontal tail.
If the aeroplane even has a horizontal tail.
The sweep angle.
The angle of incidence in the landing flare. 


When a wing enters ground effect, its lift coefficient rises: the wing creates more lift at a given speed and Angle of Attack. A wing is considered to be in ground effect if it is within about half wingspan distance to the ground. Since the main wing has a much larger span than the horizontal tail, it enters ground effect earlier, creating a nose up moment due to more lift on the main wing only.  Low wing aeroplanes have a more pronounced nose-up effect than high-wing aeroplanes do.

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An increased nose up angle brings the horizontal tail closer to the ground, which will enter ground effect as well at a certain point. Which will the counter the nose-up moment.

A wing with large sweep, at landing angle of incidence, will see the wing tip area enter ground effect earlier, creating a nose down moment.

Concorde enters ground effect with the wing trailing edge first, creating a nose-down moment. In flight this would be called Increased Stability, however in the landing a nose down moment may introduce nose wheel slam.