I know cheap construction 2x4s can warp

There are multiple possible causes for distortion in a glued-up panel like this. Often more than one of them are to blame, although there may be one major culprit. I suspect that's the case here.

Since your tabletop ended up 'in wind' (twisted) and didn't cup or bow I think the wood itself is likely to be mostly or solely to blame, with a possible contribution from the jointing of the board edges or stresses introduced during clamping (one edge tight and another looser, due to irregular clamp pressure).

If the wood when you bought it was not well dried, sadly very common in 2x material, it might have had an equilibrium moisture content (EMC) well above the norm, e.g 18% or higher. The further drying of the wood down to 10-14% could account for this, in concert with the grain pattern in individual boards which can lead to more or less of a tendency to change shape during shrinkage, see diagram below.

The only way I cab think of is to plane it a lot with my hand plane.

You can in theory plane warped glue-ups down to flat just as one does with an individual board that's in wind, but you can lose a lot of thickness this way. In this case the warping appears to be so severe that it's not a viable fix as the high corners will end up much too thin.

I suppose the best fix for this is to start again from scratch with new material (ideally selected for even, straight grain and minimal knots) but it may be possible to rectify the problem using the existing wood.

First step would be to rip it back into individual boards, sawing by hand if necessary. Then you flatten each one as much as needed to allow you to proceed and when that's done you re-joint the edges and glue the tabletop back together again. Once the glue has fully dried you'd complete your flattening of the top and then affix it to the leg/apron assembly.

There are a few techniques that can be employed to help a tabletop stay flat over time that you may wish to incorporate into the table. Firstly some general info on allowing for expansion:
What general considerations do I need to take into account for wood movement?

Hardwood buttons and expansion plates:
Will these table legs support this table?

Battens to help hold a tabletop flat:
Stabilizer for softwood tabletop that bows

I know cheap construction 2x4s can warp

There are multiple possible causes for distortion in a glued-up panel like this. Often more than one of them are to blame, although there may be one major culprit. I suspect that's the case here.

Since your tabletop ended up 'in wind' (twisted) and didn't cup or bow I think the wood itself is likely to be mostly or solely to blame, with a possible contribution from the jointing of the board edges or stresses introduced during clamping (one edge tight and another looser, due to irregular clamp pressure).

If the wood when you bought it was not well dried, sadly very common in 2x material, it might have had an equilibrium moisture content (EMC) well above the norm, e.g 18% or higher. The further drying of the wood down to 10-14% could account for this, in concert with the grain pattern in individual boards which can lead to more or less of a tendency to change shape during shrinkage, see diagram below.

The only way I cab think of is to plane it a lot with my hand plane.

You can in theory plane warped glue-ups down to flat just as one does with an individual board that's in wind, but you can lose a lot of thickness this way. In this case the warping appears to be so severe that it's not a viable fix as the high corners will end up much too thin.

I suppose the best fix for this is to start again from scratch with new material (ideally selected for even, straight grain and minimal knots) but it may be possible to rectify the problem using the existing wood.

First step would be to rip it back into individual boards, sawing by hand if necessary. Then you flatten each one as much as needed to allow you to proceed and when that's done you re-joint the edges and glue the tabletop back together again. Once the glue has fully dried you'd complete your flattening of the top and then affix it to the leg/apron assembly.

There are a few techniques that can be employed to help a tabletop stay flat over time that you may wish to incorporate into the table. Firstly some general info on allowing for expansion:
What general considerations do I need to take into account for wood movement?

Hardwood buttons and expansion plates:
Will these table legs support this table?

Battens to help hold a tabletop flat:
Stabilizer for softwood tabletop that bows