This is very confusing.
The default position is that the court system believes that you have committed two offences, failed to respond to correspondence about them including documents telling you about the allegation of criminal offences, and so the courts have convicted you of these, and issued fines which you then have failed to pay.
If the first you heard of the first case was the contact from enforcement agents seeking the payment of the fines, you should have taken advice at that point, urgently, and used the process to declare that you had not heard about the proceedings. This is called a “statutory declaration” and should be made within 21 days of becoming aware of the proceedings. Since you did not do that for the first case, and paid the outstanding fine and charges, it would be difficult to reverse this. A court may accept an out of time statutory declaration but it might be sensible to pay for a solicitor to assist with that if it is something you want to pursue. You would need to show significant reasons explaining why it is important and in the interests of justice to undo the conviction very late.
If you have only just heard of the second case then the statutory declaration process is open to you. This is nothing to do with the debt collectors though you may need information they have but haven’t given to you, including the name/address of the court which convicted you and the offence charged. You make the declaration in a semi-formal process before a court or a solicitor, where you and an official witness sign a form. Doing that - and pleading not guilty if you need to enter a plea at that point - would enable you to communicate with the prosecutor (probably a train company) to try to convince them either that this is a case of mistaken identity or that it would be more appropriate to end the criminal case in exchange for repayment of fares avoided and a fee. Pursuing the mistaken identity point is likely to require you to at the very least raise strong suspicion that you weren’t the person travelling. This will require clearer presentation of the reasons you are sure it wasn’t you than there have been in this thread. Assertions that you wouldn’t give your name and a false address are not worth anything; it’s entirely possible for someone to have a form of proof of name but no proof of address, for instance, so a railway agent would write down the name on a bank card or similar, but accept the (false) address given. Making as good a diary as you can of what journeys you made each day for the last few years would be a good start, so when you find out the date of the incident, you can account for what you did that day. If you can afford professional legal advice, and consider that worth paying to remove a criminal conviction, that may be sensible.
The conviction for the first offence should be spent under the terms of the Rehabilitation of Offenders Act and will possibly not show on DBS checks but should be declared if an employer or other organisation is not covered by the Act.
