I was late for an important meeting so jumped on intending to pay other end, where I knew there was a gateline and an excess fare window.
The rule is that you should pay at the first opportunity. Without knowing your journey itinerary, the interchange station(s) involved, the trains used, I cannot really comment on that aspect.
However I appreciate you may not wish to divulge that information, however if you want advice you may need to seek advice privately from someone with experience in this area.
I changed trains en route and thought briefly about getting a ticket there in the 5 minute connection window. But I was on my Blackberry and as I got halfway to the ticket office I turned back, to finish an urgent email. Then I got grabbed by a plain clothes inspector.
I am unable to determine if this turning back occurred at the interchange station or at your destination, but even if that is revealed it is difficult to picture it, let alone comment on it, without knowing much more information.
However what I can say is that there have been several threads posted here (you may be able to find them with a search) where passengers have described walking past an open ticket office at the end of their journey where this was an opportunity to pay, having failed to do so, and then attempting to turn back to the ticket office on apparently seeing that their exit was blocked.
If it can be established that the passenger was going to avoid payment of the journey if it was not for an inspection on exiting the station, then a prosecution is likely to succeed. In fact, if this is the case, you are lucky to only be prosecuted for a non-recordable Byelaw offence.
If you have not been in any previous trouble it is worth considering offering to settle the matter out of court.
Have you paid the fare due yet? It is not clear from your post if you have or not. If you have not, then I would give consideration to sending both the fare due, and a sum of money as compensation to settle the matter out of court. We have advised this course of action to several other members and a significant number of them have reported back to us that this was successful, however each case is unique.