UK Landlord Forum / Life

As owner of the property, is it the owners' responsibility to start paying for the council tax as soon as the tenants move out.

https://england.shelter.org.uk/professional_resources/legal/debt/council_tax_liability_after_a_tenant_moves_out

Yes, pretty much. Unless they're still in a fixed term of six months or more, when they move out, they cease to be liable for the council tax (as far as the law goes).
The tenancy agreement may make them liable to compensate the landlord in some circumstances (but that's usually more to make the landlord feel smart than to be actually enforceable).