This may not work as moving save like this doesn't always work and runs the small risks of no saves working but you can try it.
First things first turn off the cloud saves for f1 2019 if you've not already done. You can do this by right clicking f1 2019 in your steam library, then updates and then unticking the cloud on the window.
Now navigate where your current steam installation is. Should be in program files
Go to the this location yoursteamfolder/userdata/YOURID/928600. Where you id is will be some numbers that is different for everyone.
Copy all folders and files out of that folder some where else like your desktop etc. Then delete all files and folders. Keep this location open you'll need it later. This is a back up of a back up since if this goes wrong you should just be able to turn the cloud back on and you will have the season behind save like you do now but in case not you have these files.
Now go to your old hdd. Go to your old steam installation and again go to yoursteamfolder/userdata/YOURID/928600 and look in both remote and local folders. Which ever one has files in thats the one you want . If they both have files look at the dates they were created and go for the newest ones. So now copy which ever folder either remote or local from above and put it in your current steam installation that you left open earlier. Also copy the files into there as well.
Now if it was the remote file you copied you need to rename that to local.
Now launch f1 2019 and see if your saves work.
If this doesn't work and saves aren't working then turn cloud back on as explained above and you should have the season behind saves. If you still don't have any saves working you have the backup as well