As island states you can apply for a 'statutory declaration' (SD) to get the conviction quashed. Note - this does not make the matter with the railcard go away, it simply moves the entire process back to the pre-court stage. Below is the way we would expect the sequence of events to go. I have called each part of the process a 'stage', this is not official terminology but I have used it hopefully to make the matter clearer.
Stage 1 You are reported for prosecution having been found not to be in possession of a valid railcard when making a journey.
Stage 2 The train operating company (TOC) writes to you concerning the matter. At this stage you can ask them to consider an 'out of court' settlement.
Stage 3 Upon receipt of your reply to Stage 2, the TOC will decide what action to take. This will most likely be an offer of a settlement or proceeding the matter to court.
Stage 4 Depending on the decision of the TOC in Stage 3 either an offer is made or the papers goes to court.
Stage 5 If the matter reaches court and you plead guilty or are found guilty there will be a criminal conviction plus any financial sanction the magistrate decides.
In the original case, as the TOC did not hear back from you, effectively the decision was made for them to progress the matter to Stage 5. By completing the SD you will be moved back to Stage 2. If you intend to ask for an out of court settlement it is important that you engage with the TOC. Posting a draft of your response in this thread will get suggestions from forum members as to anything they would add or take out.
All TOCs appear to prefer settling out of court rather than taking the matter to court so if your aim is to avoid a criminal conviction, etc. you should take this action.