Here is a really, really hacky way (and only a high-level description) of doing this.
On the new laptop use xrandr
to increase the size of the screen to cover both monitors. If each laptop is 1920x1080 you would run:
xrandr --fb 3840x1080
If one screen is bigger, pick the biggest dimensions or you'll cut some off.
Run a VNC server on your new laptop. xllvnc
is nice and supports clipping (amongst other things). Clipping allows you to send just the portion of the screen you want to and in this case we want to send the stuff we can't see on the first screen:
x11vnc -clip 1920+1080+1920+0
Run a VNC client on the old laptop. Assuming you set the resolution correctly in step 2, just make it fullscreen and bam, you're done.
This is going to give you relatively awful performance (you can tweak this in the x11vnc command) and artefacts (from the jpeg compression) but it should just about work.
A better solution (in my mind) would be ripping the screen off the old laptop and buying a controller board for it and just plugging it into the external display port on the new laptop.
That or flog the old laptop and buy a monitor. They're not expensive.