It has come to my attention that if a student is using the Oxford VPN client despite being both being physically outside the university premises and connected to the internet via a home or private connection, they can still incur a fine from OUCS and their respective college.
That’s not quite right. Physical location is immaterial as the VPN connection effectively puts your computer on the University Network. OUCS never fines anybody. It makes an administrative charge to colleges (in the case of student offences) or departments (in the case of all staff except college-only staff) of £50. Colleges may choose to pass that on to the student as a fine and may increase it to cover their own staff costs in dealing with the incident, but that is at the college’s discretion and is not mandated by OUCS. OUCS has no role in student discipline and is simply recovering the staff costs associated with tracing the offending connection after receiving a legal notice from the copyright holder.

Just as OUCS expects Colleges and their students to abide by the University rules, OUCS in turn has to abide by the rules set by JANET, the Joint Academic Network, that provides internet connectivity to a large number of Higher Education Institutions in the UK. If the University of Oxford does not abide by the JANET rules (See then we become liable to be disconnected which would be a disaster for the academic research and teaching activities of this University.

Could you please verify whether this is true, and if so, why this is the case (despite the student not actually being in the University)?
Although the student may be using a non-University-provided connection to the Internet, once they have started up VPN and are logged in, their machine is then on the University Network and able to use all University resources, and put the University at risk of legal action or disconnection from JANET if illegal activities are carried out on it.
What is OUCS's policy on illegally downloading material from the internet?
OUCS is governed by the Regulations Relating to the use of Information Technology Facilities which you can find at Section 7.(1) is quite clear about unlawful activity. The rules are set by the ICT subcommittee of the top-level University Planning and Resource Allocation Committee. We are also bound by the JANET rules as mentioned above.
Why can both the college and OUCS fine the offending student?
We can’t and we don’t. OUCS recovers costs and is not in a position to comment on or mandate how colleges in turn recover that.
Do you think it's fair that OUCS and college rules regarding internet usage extend beyond the physical university premises?
It’s fair that the rules apply whenever people are using University facilities. VPN is a University service that is meant to be used outside of physical University premises to enable access to network resources as if the computer was on the physical University network. It puts the computer logically on the University network. This is just the same as if a student abused their Nexus email account and started sending harassing, illegal or otherwise rule-breaking messages from it. It wouldn’t matter whether they were in their college or in an internet café in Siberia – they would still be breaking the rules. The rules apply to how people use the facilities, not where they use them from.
Finally, I would also be interested in knowing approximately how many fines are distributed to students per year for illegally downloading within the Oxford network and how much these fines are (£50?)?
OUCS can only report on the number of reports handled because we don’t issue penalties.  We make an administrative charge of £50 for each case we deal with either to the college or department as described above. Statistics on numbers of incidents are included in our Annual Report, for instance

We should also make clear that these reports are not simply about downloading material. They come as a consequence of the fact that when people use some types of file sharing software, once it has started downloading material to a person’s computer it then makes that material available to the rest of the internet for upload from that computer. This is where the legal offence occurs as it means that University network (including VPN if that’s how the connection is being made) is then hosting copyrighted material for distribution for which the University does not hold the copyright or distribution rights. This is why we have to cut connections off quickly to protect the University from the risk of litigation by the owners of the copyrighted material – often large media companies or software houses. We terminate offending connections quickly not as punishment but to protect the academic activities of the rest of the University, both students and staff.