The company have four choices they can make:
1. Do nothing.
2. Agree an out of court settlement - in other words, agree not to take any further legal action against you in return for your promise of good conduct and payment of a settlement.
3. Prosecute you for a criminal offence through the criminal courts. They must do this ("lay the papers before the court") within 6 months of the incident.
4. Pursue you through the civil courts to pay the money you owe them. This is much less time pressured, and we are now seeing some railway companies do this where they do not start a criminal case within 6 months.
The experience here is that most companies, most of the time, follow a mix of options 2 and 3, threatening prosecution (3) but then being willing to agree a deal (2). There are a few examples on here of companies (especially GWR) following Option 4, before again agreeing a settlement. Once they have agreed a deal, and you have paid, the matter is closed and you face no further action.
Given your particular concerns about criminal conviction, you need to understand that it is only option 3 that would give you a criminal conviction. Being ordered by a civil court to pay a settlement is not recorded in the same way, so long (as
@jfollows has said) you pay the amounts the judge orders you to in good time.