I'm a Public-Key Infrastructure (PKI) developer, we definitely develop software for that use-case, so I can speak to the software side of your question. Complying with European law is more about how you write your corporate policies (and how you enforce them).
In a use-case like that, we would recommend that users have 3 keypairs (pairs of public / private keys):
Encryption keypair for keeping documents confidential,
Signing keypair, usually for user-login operations (called authentication) like signing emails, establishing SSL/TLS or VPN connection, etc,
Nonrepudiation keypair for signing legally-binding documents.
While it's technically possible to have a single RSA keypair do all three jobs, we recommend three different keys because there are usually different policies around how each is handled. As mentioned in this answer, nonrepudiation keys usually have the strictest policies for what you are and are not allowed to do with them. These policies define what it means for a key to be "under the sole control of the signatory". For example:
- Non-repudiation keys should never be backed up to a server.
- Non-repudiation keys should only be used when the human to whom it belongs is sitting at the keyboard (ie no auto-logins like a VPN key would do)
- Corporate policy should dictate that you should never share the password for your non-repudiation key with anyone else.
- If two people do get access to the same nonrepudiation key, corporate policy should dictate that the key be destroyed and a new one issued to the rightful person.
As I understand it, each regional government has a list of key-management policies similar to the above that you have to follow in order for your digital signatures to be considered legally-binding. These policies are meant more for the administrators of your computer newtorks rather than for software vendors.
I'm no law expert, but my guess is that for your "document uploading" scenario, you'd need the uploads to be signed by individual humans rather than by a company-wide shared key. ie "This document was uploaded by John Smith, and in verifying his signing certificate, we can verify that he is an authorized document uploader for this company".