Best Approach Methods for Signal Processing
This is a very well-written question and you are correct with regards to not really needing to know what you do with the signals.

There is always a tradeoff between frequency accuracy and temporal precision.

you have identified the shortest time-span of interest: 50ms. thus you need an FFT configuration that lets you see 50ms or quicker. 

Your highest frequency is 40Hz. Therefor you want to sample at >= 80hz (nyquist rules). For simplicity lets sample at 120Hz. 

120Hz / 256 = 0.46875 size frequency bins. You need a sample block size of 256 to work on to achieve this.

256 samples at 1200Hz takes 213.33 milliseconds to complete. That is your lag, plus you cannot determine when the tone occurred during that 213.33ms time.