New save games v1.R in old game v1.01:
They work, but you need the same folder/naming scheme, so make a folder Profile_00 and copy save games into it and rename them from Profile_00_Save_xxx.dat to just Save_xxx.dat.

Old save games v1.01 in new game v1.R:
They work, basically you make everything in reverse order as above.
1. Go to C:\Users\Robi\AppData\Roaming\THQNordic\Desperados
2. Copy the file Profiles
3. Go where you have your old save games stored and go into the folder Profile_00
4. Copy save games and rename them from Save_xxx.dat to Profile_00_Save_xxx.dat

I myself will have to Google for "Steam delete save games from cloud", as Steam now copies 2 save games from 2001 every time I go into the game, one has the new naming scheme, but neither of them is shown in the game, as you also need the old Profiles file.

EDIT:
Yeah, found the solution in first link:
https://steamcommunity.com/discussions/forum/1/360672047212137212/#c360672047212008707
You just have to go into the game, ALT+TAB and delete the save games, then they don't come back.

If you want to upload deleted save games, I did manage it after a while. I think that disabling and enabling Steam Cloud (right click on game in Library -> Properties -> Updates) did it, then I copied my save games from my local backup.

What I like about Steam is that cloud stores file dates, as Microsoft Store doesn't and you have to backup manually if you care about that. And having OCD, you care about everything.

EDIT2, 15.7.2018:
To upload deleted save games, it is a tedious process.
1. Copy save games from local backup
2. Start the game, you still won't have all the saves, only the ones from last cloud save, so exit the game
3. Copy save games from local backup the second time
4. Start the game, all the save games should be there