You should receive a letter of authority from your current landlords solicitors with the details of who you pay the rent to and from what date. Any rent you've already paid for the period will be transferred to the new owner in a completion statement.
Your new landlords will also charge you insurance (assuming you pay it) at the inception of the new ownership. Your old landlords should refund you the share from completion until the renewal date.
The opinions I give are simply my opinions and interpretations of what I have learnt, please do not act upon said opinions without consulting with a suitably qualified professional.