Set the export JPG color space to sRGB and test.
Try exporting from Lightroom with sRGB rather than Adobe RGB.
>For consistency I saved both JPGs in AdobeRGB (1998) but had the same results with sRGB
The OP already tried sRGB.
My guess is that this another bad monitor profile issue. Please recalibrate your monitor. If you don't own a calibrator and since you are on windows, go to your monitor's properties panel into the color management tab and delete any profile you see there associated with your monitor. If this resolves the issue, go out and buy a calibrator. They are an absolutely essential tool for doing digital photography on a computer.
If this doesn't help, try disabling the GPU acceleration in Preferences->performance.
Oh, another issue that could be going on is that you might have auto tone selected in Preferences-Presets or that a camera default is set. Go to Preferences->presets and disable "Apply auto tone adjustments". Also hit "reset all default Develop settings"
I've now tried all of the above and the JPG pictures exported still looks different. I did also export from Lightroom to a PSD file and it too looks different than the PSD from Photoshop or the JPG from Photoshop. But interesting when viewing the pictures in Lightroom Library the JPG and PSD pictures exported from Lightroom are identical to each other, but not to any of the other formats.
On your suggestions I'm now testing having reset all of the developed settings and am using sRGB. I did note that the sharpness of the pictures that I'm having an issue with is also different from the non-exported pictures.
Follow Jao's advice, and calibrate your monitor.
You can, as troubleshooting, and possibly as a temporary fix, set the monitor profile to sRGB. If it fixes the issue, you should ideally calibrate with a hardware calibrator, which will create a new monitor profile that accurately describes your monitor.
You seem to be on Windows 10, so here's how to set the monitor profile to sRGB:
Press the Windows key+R, enter colorcpl, and hit Enter.
Add the sRGB profile, then set it as Default profile. See screenshot (from Windows 7) below.
+1 on the monitor profile.
What I notice mostly is that there is a heavy and unpleasant yellow color cast to all these. On the second example (the jpeg export from Lr), this yellow color cast is so intense it hurts. When the screenshot is opened into Photoshop there is massive clipping to the blue channel low end, and the red channel high end. The others clip as well, although not as extreme.
All this is consistent with a bad monitor profile (which can often affect different apps and scenarios differently).
This is how the first image should look IMO:
...and not like this:
Indeed, make sure you calibrate your monitor, that is one thing to do. Second, go to Develop and check whether there are any Develop settings applied to the reimported images. Everything should be zeroed out and the tone curves should be completely linear. If not, hit reset and see what happens. If you discover some settings have been changed, there is a preset being applied upon import or a camera default applied. Check again that auto tone is not set and that the defaults have been reset and that in your import panel there is no preset selected. Lastly, there is one setting that can trigger a bug when reimporting files that had their origin in Lightroom. It doesn't happen consistently but check whether in your catalog settings, in the metadata panel the setting "Include Develop Settings in metadata inside JPEG, TIFF, PNG and PSD files" is unchecked. It can lead to Lightroom settings getting double applied if you reimport the files. This would only be something to check if you find that lots of develop settings have been applied to your image.
It was some develop settings being applied as a preset on re-import of the JPG files: clarity, vibrance, saturation and a tad of sharpening.
Thanks for your help.