https://imgur.com/47180QO

From the picture (and not being wholly familiar with lathe mechanics)
I'm not quite clear what's supposed to slide on what.

I *looks* as if the threaded screw is meant to rotate and move the
lump of metal with a groove in it across the other U-section larger
piece of metal. Given that I'm not quite clear why tapping at the
green arrow points would help, especially the ones on the LHS.

The harder I look the less I'm clear about what's meant to slide! :-)

However, one thing that's worth trying (it's often worked for me) is
to hit the seized joint so that you squeeze it, preferably with a
heavy block or bigger hammer on the opposite side to increase the
effect of the hitting.

The threaded screw turns and moves the smaller metal part along the 2
rods. The 2 rods should just push out and locked with grub screws, so in
theory tapping at the green arrows should push out the 2 metal slides.
I dont want to thump too hard in case I break it.