No traces of this block in the public Internet doesn't mean it isn't being used. It may be in use in some internal network(s) instead of a private address range (yes, that happens).
Spot on. Some of the unused blocks owned by the MoD are actually provided to a few top secret things. It wouldn't surprise me if this is the case for this as well.
"The 51.* addresses are in fact heavily used by DWP, but only internally. The best bit is this: for security reasons, there is a policy that in any communication, the leading octet of all such IP addresses must be redacted. Not like it's a matter of public record or anything. I did once toy with the idea of printing out the XKCD map of the IP4 address space, write "you are here" on it and pin it to the wall near DWP data networks teams, but I didn't think it would go down well."
Running NAT behind a routable IP address range AND using a totally different range for all servers, VPNs, and various sysadmin crap? I don't believe it.
There's no way they would be so diligent to use that IP range internally without even a tiny bit of evidence externally.
This is the government, so it's practically guaranteed that there's at least two layers of firewalls between any internal networks and the internet. If any of those layers was installed by an average IT consultancy then they'll have used NAT (because that's what you always do, right?) and hidden 51./8 completely.