Barney, plse rethink! A court setting isn't the best place

to sort out an administrative mess (as this situation is) and going to court will involve more trouble and expense for you than resolving the problem with the administrators.
Suggest the personal touch will work best. Collect up all the evidence you've got showing that you
have replied to all the official emails, they haven't answered your emails to them and that you're a perfectly innocent victim (of their maladministration).
Then ring up Transport for London, explain to the switchboard operator you need to speak to someone with the clout to sort out this hassle (ask who that person might be and what their job title and department is, make a note of the information and only then allow the switchboard operator to put you through).
If they put you through to someone who doesn't answer the phone, go back to the switchboard operator
immediately and get her / him to transfer you to someone with the right authority who will speak to you.
Make it clear
in the friendliest way you will launch an official complaint if you're not put through to someone willing and able to help and you intend to go up the management chain as far as is necessary to sort out the problem. If you have to put on the pressure, ask what procedure you should follow to make your official complaint.
When you speak to whoever can help you, explain (nicely) why Transport for London has acted wrongly and how they haven't responded to
any of the contacts you have made with them to try to sort out the problem.
Talk them through the history of what's happened and send them the emails which prove your case and prove how badly they've behaved to you. Put a read receipt request on the email and follow up if they don't get back to you when they say they will.
Make sure you get in writing from them an acknowledgement Transport for London should not have issued the fine in the first place and that they will put a stop on the fines and court case. Good luck! Linda