Not sure it's related to "Clouds sync" on GOG.
I don't have GOG (just an account), but my Series X is able to upload my last save on the "cross-progression" cloud. And the icon (when the upload is done) is slightly different than the one that you have on your PC (and as far as I know, it only concern one save, not more) :
If nothing change in 24Hrs, I suggest, if it's not already done, to contact the support (a close category) :
How do I transfer my save file to the next-gen update? — Cyberpunk 2077 | Technical Support — CD PROJEKT RED
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Thanks, I'll do that if it doesn't work