I guess you know that the result can be written as a polynomial. $$w_0+w_1{\cdot}r+\dots+w_n{\cdot}r^n.$$ So your question is how to estmate the coefficients $w_i$; this is so called "cross-sectional measures" and they can be defined for any convex body $K$.
- $w_0$ is the volume of $K$,
- $w_n$ is the volume of unit ball.
- $w_1$ is the area of the boundary of $K$, I think there is no good formula for ellipsoid, but estimates are known and you could write it as an integral.
- I would be surprised if some remaining $w_i$ can be expressed by simple formula.
If you want to write $w_i$ as an integral, check for example Burago--Zalgaller, Geometric inequalities. For example $$w_i=\mathrm{MayBeAConst}\cdot\int\limits_{\partial K} \sigma_{i-1}(k_1,k_2,\dots,k_{n-1})\\, d\mathrm{area}.$$ where $\sigma_i$ is the $i$-th elementary symmetric polynomial and $k_i$ are principle curvatures. In your case it is easy to find $k_i$...