Wood shrinks more across the grain than with it. It doesn't shrink much if any with the grain. Tables are usually designed to compensate for this by having room for the top to move, usually using something like elongated screw holes. In this case, either no one built the top to compensate for the movement or it moved more than was allowed for. Something had to give.
The fix? The typical woodworker fix would be to take it the rest of the way apart at the splits, plane the wood straight and flat at the joint(s) and re-glue. You may need to add a thin strip of matching wood in the gap to maintain (or restore) alignment of the top and all the pieces attached to it. I would also be inclined to see what could be done to compensate for more movement in the future. The downside is all this work will effectively destroy the patina of the wood.
Glenn may have better ideas. The piece is a true antique and not common. Sometimes it's better to live with the results of age than it is to try to correct them.
Rodney