Is zakat obligatory on an inheritance of unknown value that has been held for ten years and is difficult to claim, and is it valid for the mother to say that it is not subject to zakat because it is used for household expenses and does not complete a full year (hawl)?

It is your legitimate right, according to Islamic law, to receive your share of the inheritance. Neither your brother nor your mother has the right to prevent you from it, and they have no guardianship over your money as long as you are an adult and of sound mind. If the inherited money is cash and your share reaches the [minimum] nisab (definition) (threshold) and a full hawl (lunar year) has passed over it, then you must pay zakat (definition) on it. You must verify for yourself whether your share has reached the nisab and whether a hawl has passed over it.