Changing the starting time has no effect on the starting point of the frequency vector. The frequency bins are the number of sine cycles per second, with the value stored in each bin being related to the amplitude of the sine wave. If you start the time slightly later, then the number of sine cycles per second does not change.
If you start measurements slightly later or earlier then what can change is the phase of the signal.
Frequency zero for fft purposes works out as the sum of the input signals. In turn that relates to the mean of the signal. Delaying a periodic signal (or advancing it) is not going to affect the mean of the signal, exact for some small changes due to not happening to measure complete cycles.