GIMP Version: 2.8.20
Operating System: Windows
GIMP Experience: Experienced User

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When an animated xcf image made up of a number of layers is playedback and detached, is it possible in gimp to produce a display with a transparency that shows whatever is on the computer screen rather than the grey squares in the parts that are "transparent" which appear around the animation image?

Absolutely, the playback dialog is always minimized so that the background around the contour of the subject which has been cut out and removed in all layers, leaves the main subject only, remaining. However as the main subject moves around in the animation the areas that extend out of the subject image leave the grey squares in these parts rather than the screen background which would be whatever is on the screen such as a screen saved image or whatever.
It is understood that all layers are "(replace)".

If you consider that Gimp is not really a content delivery system but more a content creation system, then the reason why this doesn't happen is obvious. Exporting your animation as GIF preserves the transparent background. From there you can put the image onto another background or import it to a new image in Gimp.
I think you are trying to achieve a video editing effect where your transparent background animation is used as a layer in a video timeline.

I appreciate that.
What I'm trying to achieve is to create a sort of desktop dancer animation. I'm sure you are familiar with what I mean. The dancer pops up from the lower end of the screen and dances all around it.

I do not think it is possible just to place an animation, a gif, playing on top of the desktop.

One possibility is an animated wallpaper. There are applications to do this, probably non-free and you get what is given.

An alternative is freeware VLC video player. It has options to set a video as the desktop and to loop a video.

This a video animation with a windows application on top. (using Win7) http://i.imgur.com/jAeiub5.jpg

I remember a few customers had animations that followed the mouse cursor way back in Win98 and they were often little fairies (the animations not the customers). You need coding skills for you target OS to associate the cursor position with a randomised location for an animated gif.
Like this:
http://stackoverflow.com/questions/29735402/fluent-animation-to-follow-the-cursor

Mint KDE does that, although I find it very annoying...

Even older that that.

That is very true in both these statements; it is what Gimp is, and using my image as a layer in a video timeline. However "put the image onto another background" makes me wish the other background is whatever is on the screen.