Check with the bank officer’s to be sure that you are either on the account as a joint owner, or that your mother has designated you as the "pay-on-death" beneficiary. If either of these is true, you should get the money in the account immediately when your mother dies, without the need for it to pass through probate.
My mother absolutely refused to make a will (it's a long story), but she and I jointly owned a CD, and it was her wish that I use the CD to take care of her "final expenses." When she became ill, I went to the bank and told them I was concerned about whether I would have immediate access to the CD once my mother died. The banker checked the account and assured me that it would be no problem. And it wasn't. For your own peace of mind, I would check with the bank to make sure.