I haven't read the legislation. But a quick google suggests the lawyers are right:
The fees ban applies to all new tenancies entered into on or after 1 June [2019].If you signed a tenancy before this which included agreements to pay further fees – for example, check-out fees or tenancy renewal fees – you will still have to pay these up until 31 May 2020.
But from 1 June 2020, any term in a tenancy which requires you to pay fees will no longer be binding, so you won't need to pay them regardless of what your agreement says or when you signed it.