If the deceased has died without a will then the estate will follow the rules laid down by the Intestacy Legislation. If you die without a will you are said to have died intestate and therefore you will not apply for Probate but for Letters of Administration.
Your surviving spouse (including a separated spouse) or civil partner has the power to become your Administrator and apply for the Letters of Administration on behalf of your estate. A divorced spouse is excluded as is a co-habiting partner.
If you do not have a spouse or they have died before you then another family member may be able to act on behalf of your estate but evidence of eligibility will be required by the court.