I am an end user like you. I've never heard of an SSL certificate being automatically installed on a cPanel server, but there are always system updates.... that said...
1) Was a certificate already issued?? (https://certs.godaddy.com/cert)
2) You mentioned about the private key - which maybe part of the issue and we may need to rekey the certificate.
So Step 1) In cPanel I would generate a new CSR - the private key should be on that screen as well
Step 2) In a new browser I would go to the certs page (link above) and rekey the certificate with the new CSR
Step 3) You should be able to download the certificate - I find it can take 10-30 minutes for the certificate to be re-issued.
Step 4) in cPanel (in a new tab) you should be able to upload the certificate and copy the private key from the original tab
Step 5) Make sure to then assign / activate the certificate in cPanel
Other than the 10-30 minutes for the certificate to be re-issued as you said this should be a matter of minutes
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