If you switch part way through a billing month, your old energy supplier won't charge you for the energy you're paying your new supplier for.
You will never be charged for the same energy twice. Suppliers will refund you, or, ask for any debit outstanding by settling your balance with a final bill when you move away from them.
Final bills typically take 6 weeks to be generated. If you do find it takes longer, see our article on whether you could be owed compensation.