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This paper is concerned with the estimation of spreading speed of a nonmonotone equation, which involves time delay and nonlocal dispersal. Due to the time delay, this equation does not generate monotone semiflows when the positive initial value is given. By constructing an auxiliary monotone equation, we obtain the spreading speed when the initial value admits nonempty compact support. Moreover, by passing to a limit function, we confirm the existence of traveling wave solutions if the wave speed equals to the spreading speed, which states the minimal wave speed of traveling wave solutions and improves the known results.