FFT of resultant of signals
First, you should note that the FFT refers to an algorithm for the fast computation of the DFT taking advantage of its symmetry. 

The signals you have are:



where $\mathbf{a}_y$ and $\mathbf{a}_x$ can both have negative a positive values depending on the direction along the $x$ and $y$-axis. These signals are the output of your accelerometers.

What you call "a piece-wise resultant acceleration", calculated like 
$$r(i) = \sqrt{x(i)^2+y(i)^2},$$ is just $\rVert \mathbf a\lVert$, the magnitude of a vector quantity $\mathbf{a}$, which doesn't show the direction.

From NASA webpage, about vector components:


  Why go to all this trouble? Because, in aerospace, we are often
  dealing with forces and forces are vectors. Breaking a single vector
  force into several components allows us to study the resulting motion
  much more easily.


In conclusion, performing a DFT on your vector quantities $\mathbf{a}_x$ and $\mathbf{a}_y$ will give you frequency information both for about magnitude and the direction together, while performing a DFT on your scalar vector $\rVert \mathbf a\lVert$ will just give you the frequency domain of the magnitude, so yes, you are loosing information here.