Frankly I find this situation appalling. IIRC you are a student and I would think at this time of year especially you have better things to do than be messed about like this.
Notwithstanding the clear advice from
@Puffing Devil about what you must surely now have to do to get this removed.
But
I would be interested in experts views on what aspects of formal complaints procedure could be used now (in addition to resolving the legal issue / conviction).
If I have it correct in this summary
1) the OP was not stopped for any ticketing issue by NEXUS
2) NEXUS accused the OP of a ticketing offence based on someone else giving NEXUS the OPs details
3) The OP wrote to NEXUS stating it was not him (involving time and cost)
4) The OP provided supporting documentation with his own ID / Photo that NEXUS could have used to check against their records of the person stopped (?)
5) Nexus persisted on takin the matter to court - taking up the OP's time and effort and no doubt costs
6) Nexus solicitor at court accepted the case did not involve the OP because they did not match the description held on file from the incident (something that could have been done at point 3 or 4),and verbally said they would drop the case
7) Nexus then actually failed to drop the case (presumably)
8) the OP has been convicted of something they did not do.
9) The OP has now got to spend more time, trouble and money to sort this out
In parallel to unwinding the conviction I would assume that the OP could now
A) invoke some sort of formal complaints process with NEXUS and seek redress. I am not sure what the going rate is but I'd be tempted to request at a reasonable estimate of the time involved a payment of £20 per hour + costs of postage, travel to court etc in compensation at least
B) if NEXUS don't offer to pay redress then I would escalate to an Ombudsman
c) I believe Nexus is a local authority related body (see link below) and therefore in addition to Passenger Focus it occurs to me to suggest the Local Government Ombudsman (LGO) may be an appropriate organisation to escalate this to. My experience of the LGO is that they are relatively effective, and a proper government funded Ombudsman - so should be better than the Rail Ombudsman which gets a poor rep on this forum for not being effective
How we engage with passengers and service users
www.nexus.org.uk
At this stage (or perhaps as soon as you have done the stuff Puffing Devil outlines to get things corrected) I'd be tempted to make an official complaint to NEXUS, your MP with a covering note asking your MP to pursued the matter on your behalf by also contacting NEXUS to ask them to do the work necessary to correct their error, or if they are unable to do that to pay your costs in doing so - frankly they should pay you costs to engage a solicitor to do it in my view....
If NEXUS response to you (not necessarily the MP) is inadequate, I'd escalate it to Passenger Focus and Local Government Ombudsman - assuming the LGO accepts that combined local authority bodies fall under there jurisdiction. If they don't fall under the LGO jurisdiction then I would seek clarification from the LGO which public ombudsman they are accountable to. My understanding is that there is an Ombudsman for the vast majority if not all public bodies.
I hope this advice is of use - I'd of course welcome other people's corrections or thoughts on such a course of action.
I'm sure forum members would help check drafts of complaints letters to the various bodies I mention above if the OP wishes to pursue this in this way.