Here's the verdict: Looking at [Teds 3 sunflower] photos in both Firefox and Safari on both computers at the same time and with the desktop in sRGB emulation - there is a difference on the laptop but perhaps not huge like it would be if the flower was red.
On the desktop-no difference between the three in Firefox but there is a slight difference between Safari and Firefox- the colors are very slightly more saturated in Safari. It is my understanding that Firefox has better color management than Safari but not sure about that - I do see a small difference though.
Slight differences are OK and can be expected.
It looks like there is nothing wrong with the desktop computer color management - I'm betting that the NEC could have even been left in wide-gamut mode and those three sunflower images would have still looked OK on the NEC.
On the laptop-in Firefox - sRGB is more saturated (but not to the point that I would say OMG it is just more saturated and if I wasn't doing a comparison I might think it was fine). There are some pinks or reds in the background that show more color too. Those pinks/reds are not terribly saturated but that background color does not show up on the desktop with as much strength. The flower listed as aRGB is less saturated than the previous one but more saturated than the final one, ProPhoto. The ProPhoto looks closest to the photos on the desktop. In Safari on the laptop not a huge difference between the two browsers.
So back to the laptop, what is described seems firstly symptomatic of no color-management in that the embedded profiles are being ignored and the laptop is displaying each images' data via some unknown but constant profile. That is why the saturation gets less when viewing in order of sRGB, aRGB, ProPhoto. Secondly, however, of the three, sRGB is the one that should look right with no color-management but an otherwise correct system monitor profile.
My gut feeling is that, if the laptop could be reverted somehow to it's original default profile, all would come right as rain :-)
So apparently she is still having a problem as well but has found a work around that satisfies her. I don't particularly want to have to edit in sRGB and go through the steps she is using since I am using Lightroom primarily and by doing that, as has been discussed, I would then be editing to satisfy the laptop, right?
Right! Is the laptop essential to your work-flow, may I ask?