This is a question where it is important that we are careful with our definitions. It is not possible to pay interest on savings account, but only on loans. And then the interest is payed at the end of the term, and it is payed because you have had the option to do something with what you have lent. So the question would rather be: when did banks start to claim that they are paying interest on deposits (what you are calling savings account.). In essence what is happened is that there have been a mix-up of two different type of banking transactions, the irregular deposit contract and the monetary loan contract.

The practice of confusing these two came about as a consequence of the canonical prohibition on usury. To avoid the ban lenders began to camoflauge their loan transactions as deposits, called depositum confessatum. It was in name a deposit contract, but in practice it was a loan contract. When the term of the deposit reached its end and the “depositor” “failed” to return the deposit, he was forced to also pay a pentaly, i.e interest. With the beginning of this practice, which must have started after Charlemagne prohibited usury. After this we have a gradual evolution until our time. It seems reasonable to assume that the practice of banks claiming that depositers pay fee, and not “earn interest”, ended completely during the 19th century. This century marked the death of 100% reserve banking policy within the Bank of Amsterdam. It would not be possible for a bank that followed the policy of Bank of Amsterdam, which kept almost all money people had deposited to them in their vaults, to pay “interest” on deposits.

Sources: Money, Bank Credit, and Economic Cycles by Jesús Huerta de Soto