There is hardly any momentum because the thing is barely moving. Momentum depends on mass and speed, and speed is basically zero.
I guess what you’re really asking why it doesn’t slam down when there is presumably something pushing it very hard. But the thing isn’t pushing the way you think it is. It’s pushed by increasing fluid pressure. When the thing preventing motion is removed or breaks, the press will only move as far as it can before that pressure is equalized with (something, don’t really know what; either atmosphere or some internal reference pressure I guess). There will be some overshoot and rebound but the distance will be very small, because even a small movement of the press will result in a huge change in pressure.
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