to add to the helpful advice already given here is a copy of general advice often given on this forum which may be of help to you to give you an idea of the process and what you should expect as things progress.
I'm afraid any saving you hoped to make by combining a train and coach journey is going to be wiped out by your decision to stay on the train without a ticket and then try to dodge the fare when you arrived at Paddington with a 'short ticket' - had you actually bought a ticket from Swindon whilst on the train (but before the train stopped at Swindon) you would have been in the clear as the ticket would have been valid.
anyway - here's the general advice for you:
Firstly, you committed a criminal offence by travelling with an invalid ticket and railway companies are entitled to prosecute you in the Magistrates Court if they want to. The good news is they will normally offer an out of court settlement to people who co-operate with them and who haven't come to their attention before.
Next, what happens is the railway company will write to you. The letter will go to the address that you gave them / on any of your ID they checked so you need to be able to access hard copy post at that address as they often send things that way (but also check e-mail and spam). It is extremely important that you reply, and you'll only have a limited time to do so (often 14 days from the date on their communication).
When you receive the letter post a redacted copy of it in this thread, along with your draft reply and forum members will proof read it for you. You'll find lots of example letters you can use in this section of the forum but the reply basically needs to mention:
- That you are sorry for what has happened
- What you have learned from the incident and will not do this again
- That you are keen to settle the matter without the need for court action
- Offer to pay the outstanding fare and the railway company’s administrative costs in dealing with the matter
The railway will probably offer you an out of court settlement. Expect to have to pay the cost of the fare for the journey you made at the full Anytime rate with no credit given for the invalid ticket you purchased*. In addition they will charge an admin fee, typically £150. They may / will also audit your online ticket purchase history and if this shows you've purchased other tickets that were not valid (eg railcard discounted tickets when you did not have a valid railcard, or tickets for only part of the journey, or tickets bought after boarding the train you travelled on ) they will probably factor the cost of these journeys into the settlement offer.
Settlements usually have to be paid in full within a few days of them being offered so if money is tight you might want to think about how you will raise the sum required at short notice, or take steps to save it up between now and when they write to you.
* The Reading to Paddington ticket wasn't valid because you only bought it on arrival at Paddington, not before the train stopped at Reading
I'm afraid an Anytime Day Single from Swindon to London is £87.90 so that is the fare they are likely to ask you for (had you bought before train stopped at Swindon I suspect you could have got a 'Super Off Peak Single' at £36.70 (further discounted for your Railcard which likely now won't happen either) , if the time of the day was suitable. You might need to budget for c£200+ to resolve all this if you are offered an out of court settlement.