The coroner will often issue an interim death certificate to enable probate to be applied for. Indeed, you don't actually need to submit a death certificate to apply for probate.
As a point of practice, the estate is not completed and passed to the probate office. What happens is that the personal representatives complete an oath for probate or letters of administration detailing how they are entitled to the Grant of Probate/Letters of Administration together with an IHT form. Once the grant has issued the personal representatives can deal with the estate.