When I would click to check for update it would show there was a newer version and I would click to update now and it would go thru all the motions like it had done it but the version would stay the same. Window10 Pro 64 Bit v5.8.5308 was where I was at when I went thru the motions to update - five times with a reboot of system and it still would not update - I solved the issue buy going and downloading the CCleaner Pro trial and clicked install and it installed on my computer and updated my CCPro to v5.10.5376 (64-bit) Just strange may be it the new Browser Edge
Years ago someone had an issue with CCPro not updating automatically before because of the System/Windows Certificates which can be updated via Microsoft Update. After updating the certificates it fixed the issue, although I don't know if it will or won't in your case being Win10 is so new.
If that doesn't work and since your a registered user of CCPro you're entitled to get help directly from Piriform at: