What are the units of my data after an FFT?
In short: if you directly use the voltage output, your FFT amplitude should be in mV as well. If you have the sensor calibration curve, the FFT amplitude should be in Teslas per second (T/s), as you look at a derivative.
If you look at a power density spectrum (squared), then the units above should be squared as well (mV$^2$ or T$^2$/s$^2$).
In general, for a given  unit U, albeit in volt (V), tesla (T), whatever. The FFT sums samples in the original units (U) multiplied by unitless complex values (due to discretization). The units after FFT remain the same as for the original signal, i.e. U.
If you take the absolute value, the same again. For instance, the $0$-frequency index gives you the DC, or average value of your signal  (or at least it is proportional, with a factor of the number of samples, or its square root depending on the normalization).
The continuous Fourier transform, in contrast, &quot;sums&quot; the samples times complex cisoids times a time differential (the $\mathrm{d}t$). Its units would be U.s (s for seconds). Unless you normalize it by something with a unit.