Variance of an Implicit Function of Kalman State Vector
I think I have the solution.
I'd be happy to hear others' thought.

Defining $ F \left(r, v, a, {T}_{tth} \right) = r + v {T}_{tth} + \frac{a {{T}_{tth}}^{2}}{2} $ which is the implicit function which connects all variables.
Since we're dealing with non linear function the variance is given by:

$$ var \left( {T}_{tth} \right) = J P {J}^{T} $$

Where $ P $ is the state vector covariance at a given time and $ J = \left[ \frac{\partial {T}_{tth} }{\partial r} \frac{\partial {T}_{tth} }{\partial v} \frac{\partial {T}_{tth} }{\partial a} \right] $ 

Using Total Derivative Law on the Implicit Function:
$$ \frac{\partial {T}_{tth} }{\partial r} = -\frac{{F}_{r}}{{F}_{{T}_{tth}}}, \frac{\partial {T}_{tth} }{\partial v} = -\frac{{F}_{v}}{{F}_{{T}_{tth}}}, \frac{\partial {T}_{tth} }{\partial a} = -\frac{{F}_{a}}{{F}_{{T}_{tth}}} $$

Where $ {F}_{x} $ is the partial derivative of $ F $ with respect to $ x $.

Each individual component would be:

$$
\frac{\partial {T}_{tth} }{\partial r}=-\frac{1}{v+aT_{tth}}, \frac{\partial {T}_{tth} }{\partial v}=-\frac{T_{tth}}{v+aT_{tth}}, \frac{\partial {T}_{tth} }{\partial a}=-\frac{T^2_{tth}}{2\left(v+aT_{tth}\right)}
$$

Transform the state covariance matrix with $J P {J}^{T}$ and you'd have the right solution.

Given a diagonal state covariance matrix, we should have:

$$
P_{tth}=\frac{\sigma^2_r}{\left(v+aT_{tth}\right)^2}+\frac{T^2_{tth}\sigma^2_v}{\left(v+aT_{tth}\right)^2}+\frac{T^4_{tth}\sigma^2_a}{4\left(v+aT_{tth}\right)^2}
$$