As you say it will be handed to the council, I assume this is an older persons or disabled persons bus pass, issued under the English National Concessionary Travel Scheme (ENCTS), which are a standard format across the country. Not knowing what your own pass looks like, or if it's in a wallet or anything like that, you could have some explaining to do.
In the first instance, you'll need to tell your mum, as I strongly suspect she'll need to take it from there in regard to getting it returned or reissued. I would advise that she contact the issuing local authority (be that Leicester City or Leicestershire), and explain what happened. It's not something we see a lot here, but it may help if you could provide them with evidence that you have your own pass (especially if it's also a white card with a photo and a logo on, or in a similar wallet). They may be sympathetic, they may not be.
As for you, your dealings will be with the bus company. Again, if you can present them with evidence that you have your own pass, fully paid for, you may be able to get the pass returned before the council are involved - it depends how quickly they process them... it may not even be back in the depot yet, depending on how often the inspector goes back). You were travelling without a valid ticket so your penalty fare was issued correctly, but speaking to them might cut off further steps.
The downside is that bus companies tend to have centralised (i.e. nationwide) customer service centres, so if there is any contact details on the back of any paperwork, use those as they will go to the right department.
