Right then, that looks like you have been fined in accordance with a guilty plea and no financial information known about you (this assumes an income of £440pw). This would be half a week's income (£220) reduced by a third for your guilty plea.
As noted, you are out of time for an appeal to a higher court and you are unlikely to be given permission to launch a late appeal. That said, you could only appeal against the sentence anyway as you pleaded guilty, and that would only be worthwhile if your income is below £440pw. I believe your only avenue of approach would be to have the matter re-opened under S142 of the Magistrates' Court Act. This says:
A magistrates’ court may vary or rescind a sentence or other order imposed or made by it when dealing with an offender if it appears to the court to be in the interests of justice to do so
You should request this in the basis that (presumably) you were not given the opportunity to provide details of your means and you would like the matter revisited. One other thing worth noting is that I believe (though am not 100% sure - I'll try to look it up) that the "Victim Surcharge" is not applicable where a conviction under a railway bylaw has been recorded. If that turns out to be true then that's at least £32 you should have knocked off.
One thing you must now bear in mind - a court has recorded a conviction against you and only a court can set that conviction aside or make any amendments to the fines and costs imposed. Whilst it might help to engage with the railway company (see below) the matter is now entirely outside their control. Even if they now decide that they should not normally have taken court action against you, they did and that action has seen you convicted. This is what I believe you should now do:
1. Contact the Justices' Clerk at Nottingham Magistrates' Court and make a request to have your case re-opened under S142. The grounds would be the level of fine (because you did not provide details of your means) and possibly the Victim Surcharge (I'll try to find out the answer). You will probably have to appear in court to do this (though different arrangements may be in place due to Covid) and you should be given a hearing date.
Beware with this. These matters are usually handled by non-legally trained people. You may well get the reply "you pleaded guilty so the case cannot be re-opened". That is incorrect and you must be prepared to stand your ground if you receive such a reply.
2. Contact the fines office at the same court and let them know you are making the above request. They should agree to postpone any enforcement action until the outcome is known.
This is important. If you don't do it you may have bailiffs knocking on your door.
3. In the meantime you can contact the rail company as suggested. As I said, they have no control over the court conviction but you may get some information from them that might lead the court to take a different view. The court actually has the power to set the conviction aside but this is a big ask and it will depend entirely on what you find out from the railway company.
Finally:
What I want is to just pay the original £20 fine. More than happy to do so. I admit that I didn't have my ticket on leaving the station and that's that. No issues with the fine. I appealed the next day, they read me the garb and said they would contact me. No one did.
I want the court element of all this to be dropped and to pay the fine issued by the train company in the first instance
Do not get overly hopeful of that outcome