A one sided test like a $t$-test can be obtained by merging your two separate regressions into one that models both groups. If you are interested in the effect of x on y across groups then add a $x_i D_i$ regressor where $D_i$ is a dummy variable that codes your group:
$y_i = a x_i + b x_i D_i + u_i $
As you can see, your group effects are given by $a$ and $a+b$. Thus your hypothesis test becomes $H_0: b>0 $ and its statistic would be based on a one sided t-statistic
These 2 IV are not going to be collinear as there is no linear combination that makes its sum = 0 under standard conditions (and a OLS, ML estimator). To see why let's only take the sample for which $D_i=0$ and assume the following vector notation:
$y = x a + z b + u$
where z is a vector with $x_iD_i$components.
Collinearity may airse as the linear combination $c_1x + c_2z$ approaches zero (for $c_j \neq 0 $). But just considering the block of observations where $D_i=0$ automatically leads to a linear combination block equal to $c_1 x\neq 0 $ if $x \neq 0$.
I guess you are thinking in term of standard theory that tells you that higher IV correlation may lead to collinearity, but don't forget that standard theory refers to the IV which in our case are $x_i$ and $z_i(=x_iD_i)$ and not $x_i$ and $D_i$. Moreover, the mathematical issue behind the correlation issue is the fact of having linearly dependent variables, so linear dependence between $x_i$ and $z_i$ must be addressed.