Here's a suggestion. They might appreciate it if you write with the full story now, rather than letting them work on the case under the impression either that it was your husband all the time or that there is some doubt over who used the card for the earlier period. That might apply even if the work they do is fairly negligible - if you don't tell them, someone in the office might feel that they are unnecessarily sending paperwork out again. Also, if you don't tell them they won't know that you are willing to admit it at this early stage; they might think that you kept on wanting to keep it secret for several months.
He didn't mention my name
It may not be easy for your husband to know how the staff member perceived the issue of who used the card earlier. Did they just say something like "it looks as though similar journeys were made last year" or did they take anything he said or did as indicating it was him? Even if it doesn't matter legally, it may be a factor in whether you write to them as above.
how many months/years this conviction is going to be under my name?
The basic rule is that it depends on the sentence. As a Regulation of Railways Act or byelaw conviction would result in a fine, it would be "spent" after a year for anyone over 18 years of age.
https://www.gov.uk/exoffenders-and-employment
(The page is wrong about the period for under-18s, which is in fact halved.)
A Regulation of Railways Act conviction, being for a "recordable" offence, would be put on the Police National Computer and so automatically appear in DBS checks, but still "spent" after a year. A byelaw conviction is unlikely to be disclosed on a DBS checks because even if the police force is aware of it they may be likely to think it isn't relevant. That is a separate issue from your duty to disclose it if asked. For some jobs and perhaps for some professional organisations even spent convictions would have to be disclosed, and higher-level DBS checks would reveal spent convictions for "recordable" offences.
You're writing about yourself in the singular. Whether they would regard this more seriously because you both used the card and so might reasonably be thought to have communicated about wrongful use of it may be something a solicitor would be best qualified to comment on: that might be an additional reason to that given by
@furlong above for consulting one.
If either of you have union membership, or relevant benefits which come with a bank account or insurance, you may be able to get free legal advice. Otherwise the Law Society has a register of solicitors.
https://solicitors.lawsociety.org.uk
Other sources of information and advice are these, both of which have helplines:
https://www.nacro.org.uk/
https://hub.unlock.org.uk/information/