It's your responsibility to notify the utility companies when you move in. You should also take meter readings when you move in. It's too late for this now but you should certainly take a reading when you move out and keep a record of this. It would be worth taking a meter reading now as well so that you will at least have a rough idea about how many units you use each day.
As for what else you can do now, in your shoes, I would simply write "no longer at this address" on a few of the unopened envelopes, put them in a post box, and leave it to the electricity company to sort something out. It's worth putting a few pounds on one side in case they decide to send you a bill but if you hear nothing in the next 12 months, you should be thankful to the previous tenant for not caring about his money and paying the bills.
You should be sure to take meter readings and notify suppliers when you move in to the next property.