Transmission of shares in Demat account on basis of Registered Will

Probate Order' herein means  copy of a Will certified by Court of competent jurisdiction, which is to be treated as a direct evidence of the authenticity of a will thereby granting administration to the assetSection 213 (1) of the Indian Succession Act, 1925, inter alia states that a legatee’s (a person who inherits under a will) right to property bequeathed in a will cannot be established in a court, unless a court of competent jurisdiction in India has granted a probate of the will under which the right is claimed or letters of administration with the will annexed thereto have been obtained.

However in certain states obtaining probate is not compulsory, since your query is silent on the fact in which state the property is situated ,we will not be able answer, whether you required probate.
Also, if in the bank accounts, you are the nominee then the bank should not require the copy of will. If they are adamant then you will have to apply for probate.

With respect to your query "Whether you should get your will probated, depends on the fact of your case. If  there is no dispute between the legal heir and Will is not ambiguous , then there is no need for probation. You can get the property mutated in records of MCD and other authorities like electricity board and Jal board etc. If there is a dispute, then you can file an probate application .