It will be best to contact the lender and give as much information as possible. Many application forms let you give both your current address and recent addresses.
Before moving out, you should contact the council and utility companies (water, gas, electricity, phone, etc) to tell them about the move and to provide final meter readings. It's possible that there will be a repayment due to you or you might owe some money. Either way, you should give your next address.
You should only pay what you owe. I'm sure that the council will be happy for you to keep paying at the current address but, unless you tell them about the move, they may ask you to pay at both old and new addresses.
If there will be anything important coming in the post, you can contact Royal Mail and (for a fee) they will forward your mail.