It’s quite possible your local council will let you do this online - mine does, it takes a couple of minutes to advise the council of move out\move in dates. Once done, the council should reissue the bill in the tenant’s name at the correct time; after that it’s all between the tenant & the council (eg. I’ve had mature students move in before, so they were paying £0, whilst my council doesn’t give the slightest discount for fully furnished empty property so I was paying full rate during the void period)