There is no such thing as a reading of the Will. This happens only in films. The normal procedure is to apply for probate.
However, if the net estate value is less than �5,000, you can administer the estate without a Grant of Probate. Under the Administration of Estates (Small Payments) Act 1965, assets that are held in a bank or building society can be paid to the personal representative named in the Will on receipt of an indemnity from him or her. However, if your nan owned any shares a Grant will be necessary.
I assume your mother's name was on the bank and building society accounts for her to assist your nan (rather than as co-owner). In that case, strictly speaking you should follow the procedure above. However, if the estate is under �5k, I can't see a problem with your mother doing as you suggest.