Is a deposit, for a term of ten years, with a value of ten thousand pounds to become thirty thousand pounds in an Islamic bank, permissible, knowing that Zakat is paid every year on the original amount?

Whoever deposits money in a bank with a known increase after a period of time has fallen into prohibited Riba (usury/interest). This is not a permissible investment. For an investment to be permissible (halal (definition)), it is a condition that the principal is not guaranteed, that the profit is a percentage and not a known fixed amount, and that the field of investment is permissible. Your mother must repent and reclaim only the principal without the increase. She must also know that her paying Zakat (definition) on the principal does not absolve her of the sin of Riba.