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I'm afraid that even if there is a Will, Probate is required to be obtained, IF your Mother left an Estate valued at over £5,000. Some Banks will allow the Executors to close the deceased's account if ther eis under £5 or £10,000 in it without seeing Probate, but the majority will insist that you both obtain Probate if the value of the Estate is over £5,000.
What I suggest you do is to speak to the Bank concerned and see what their policy is.
I hope this helps.