In the UK research shows that around 60% of adults have not written a Will.

If you die without leaving a Will or the Will is invalid you will have died 'intestate'.

The rules which govern this in England and Wales are set out in the Administration of Estates Act 1925 and the Inheritance and Trustees Power Act 2014. In addition, individuals may have a claim against the Estate under the Inheritance (Provision for Family and Dependants) Act 1975, if such as claim is made it would make the probate contentious and the contentious matter would need to be dealt with a by a specialist lawyer to resolve the claim, this is not something we deal with.

There is a Government guide Intestacy - Who inherits if someone dies without a will

Who can Administer the Estate for Probate?

Normally it would be the Next of Kin - Spouse, Civil Partner or Child

You can not apply is you weren't married, there are no automatic rights for co-habitees.