It depends on what their assets were as to whether they need Letters of administration or not.
You have to wait for the death certificate, generally, which can take a while. Then an assessment needs to be made of the assets and information gathered for the application. At the earliest, you might be able to get an application into the Court within 2 or 3 months.
The children are likely to be the most eligible for the grant, but that will also be a matter of assessment.
If only one child applies, then the others would generally have to provide consent to the court that the one child act, because they all have the same standing.
The application for letters of administration doesn't have to be done with the assistance of a lawyer, although it makes it far easier. Likewise with the administration of the estate after the grant.