Does zakat become obligatory on money saved in a usurious bank—due to the absence of an Islamic bank—for the purpose of purchasing a necessary commodity, knowing that the saver is in need of this money?

If you are compelled to deposit money in an interest-based bank for safekeeping, there is no sin, but do not place it in a savings account; only use a current account. You are obligated to pay zakat (definition) on your capital deposited in the bank every year if a full year has passed [since its acquisition] and it reaches the nisab (definition), whether it is saved for buying a car or a house. Money added during the year is subject to zakat along with it, or it can have its own independent hawl (zakat year).