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A miracle has happened. I used Heroes on two computers where I had the synchronization issues in the cloud as described above.
I installed Heroes on a brand new computer, of course with my own account, and here's the miracle: the saves from the last computer I played on were loaded. And then they loaded without any problems on both other computers.
It seems that the problem was considered in technical support based on the game error reports that were sent. I don't think the new computer is the reason. I thought that very old saves would be loaded onto it. But no!
This is the second time this has happened; the first time was with the problem of the game not launching in Uplay, and then the problem suddenly disappeared.
Respect! What more can I say?