I did two Win8.1 to 10 upgrades, one of which dual booted with Ubuntu, and both went seamlessly using the Win10 ISO (I didn't boot it, but mounted it as a Windows drive and ran setup.exe). My one upgrade from Win7 was awful, with repeated failures like yours (though not the same message), and I tried so many things that I cannot now remember what eventually worked. If you're not using the ISO, maybe you could give it a try.

@AFH : I tried the iso and got exactly the same issue. I also tried through Microsoft update, in which case it fails without any specific error message. Because I backed up everything sector by sector, crashing isn't a real problem (though restoring would take 4 hours). You probably got the same error message since my translation come from mine as I couldn’t found the original English message.

I don't think it was the same message, because I got an error code which eventually led me to a solution. Your translation is good, but you would need to know the exact equivalent message (which may not be a literal translation) for a search, though you may just strike lucky with the French language version. Apart from that I don't know what to suggest.

@AFH the search on the french version lead to nothing. There should be more and better messages with the original english. But is their really no way to bypass compatibility checks?

I searched for "Un problème est survenu." "Désole, nous ne pouvons pas déterminer si votre PC peut exécuter Windows 10." and one of the links was headed "Windows 10 update : error 8007001F - Microsoft", so I searched for this and got a few more links in English.

If the error isn't spurious, are you sure your system meets the Win10 specifications? If not, I don't know how to bypass the check, but others may be able to help.

@AFH read again the hardware details I wrote in my question. I have also 46Gb of continous unpartioned space. The point is one of those checks (I'd like to know which one) fails to work whereas it should return ok and let the upgrade process to continue.

The hardware certainly seems OK. Have you run the compatibility check manually? It should tell you what it is objecting to. If it's software, you may be able to patch the installation details in the registry to hide the problem software, then patch back afterwards. If you need software which won't run, you still have the option to revert to Win8.1 (unless you use a pre-1511 ISO and then upgrade to 1511, which replaces the Windows.old directory containing the Win8.1 files).

@AFH : Ok, but in order to do this, how to download and re‑install KB3035583 manually (it seems I tried too much things and KB3035583 no longer appear in Microsoft update) ? Otherwise, yes it’s 1511.

After much chasing I found this link to down-load KB3035583 directly. If this doesn't help we may need to move to chat.

OK, but I'm just going out for a couple of hours.

@AFH the dism command on the .cab failed with Error in operation: the package is not applicable.

with error code 0x800f081e