(More a comment that is too long for the comment box) ...  Given a sample of size $n$ from a standard Uniform parent, one can show, by induction or otherwise, that the joint pdf of the first $k$ order statistics, is:

 $$f(x_{(1)}, x_{(2)}, \dots, x_{(k)}) = \frac{n!}{(n-k)!}\left(1-x_{(k)}\right)^{n-k} \quad  \quad \text{for }\quad  0 < x_{(1)} < x_{(2)} < \dots <x_{(k)} < 1$$

... which has a neat functional form.

Your problem then reduces to:

> Find the cdf of the sum $X_{(1)} + X_{(2)} + \dots + X_{(k)}$ given joint pdf $f(\centerdot)$ 

... which unfortunately will result in a $k$-part piecewise solution with kinks at $(1, 2, \dots, k)$. 

Using the _mathStatica_ package for  _Mathematica_, I obtained the same solution you provided for $k=2$, and was also able to derive a general exact solution for the sum in the $k = 3$ case (in 3 parts). However, an exact general solution seems to get messy quite rapidly.