As
@Bod says, once you have got the certificate of deputyship from the COP, book an appointment to see the bank manager at your mums bank (or a branch of her bank) and tell them it for COP deputyship. You will be asked to bring the COP certificate and some ID for yourself - exactly what varies from bank to bank, so do ask what you will need.
They will register the deputyship with the bank and you will get a cheque book and credit card with your name on it stating that you are the deputy for [your mums name]. You should receive then within a week or two. The account number etc will remain the same. Bank statements will be sent to your address and you should be able to open an on-line account to manage her account, if you wish.
Once you start using her account do keep all receipts as you are likely to need them later when filling in the annual financial report.
You will probably get a phone call from someone in the Office of Public Guardians to see how you are doing, guide you through problems and answer your queries, but you do not have to wait for this.