First part: If your deceased sister left a will then the provisions of that will must take effect. If she didn't then the laws on intestacy apply:
https://www.kingstonsmith.co.uk/wp-content/uploads/2016/06/Intestacy-Rules-flowchart-if-you-live-in-england-1.pdf
(So, unless your sister had a surviving spouse, her estate is shared between her children)
Second part: Where a will leaves something to a child of the testator, but that child pre-deceases the testator, then whatever would have passed to that child passes to the grandchildren who were parented by that child.
So your mother needs to nothing to alter her will.