It happens to the best of us. It's frustrating when you've missed your train, but what you're able to do next depends on the type of ticket that you have, and why you've missed the train.
Missed train due to rail delays
If you've missed the train because of another rail delay, then you're covered. It doesn't matter if you're using separate tickets, either. So long as you've left enough time to change trains, then it's no problem that you've missed the train.
Do you have a flexible train ticket, such as an Off-Peak or Super Off-Peak ticket? No problem. Just hop on the next train, so long as your ticket doesn't have any restrictions on which company you can use. In this case, you might need to stick to the operator on the ticket.
If you have an Advance ticket, you can get the next train operated by the same company that's specified on your ticket. If there's widespread disruption, or a very long wait until the next train (over 60 minutes), then you should be accommodated on a different company at no extra cost. You can find out more about your rights in the National Rail Conditions of Travel.
You will be offered First Class compensation if you possess a First Class ticket and there is no First Class on the next service. Claims can be made through the train operating company you are travelling with.
Missed train due to external delays
Have you missed your train because of an external reason, such as an overrunning meeting or traffic congestion? If you have a flexible ticket, that's no problem. Just make sure that your ticket is still valid for the time of train that you want (eg. if you have an Off-Peak ticket and it's near peak time).
If you have an Advance ticket, speak to staff if you can. Officially, you would need to buy a new ticket if you're missed your train. However, if there's been an accident on the roads nearby, or Tube and Metro delays, then staff can show discretion. They might even be briefed to accept tickets if it's affecting a lot of customers.
Don't board a train in these circumstances without speaking to staff first. It's highly likely that you'll be asked to buy a new ticket, or face a Penalty Fare.