The short answer is, that the total force between the streams is the same in all inertial frames. In the moving frame where the electrons are at rest, the force is repellant and therefore, the force will be repellant in all inertial frames.
To see this, consider a dense stream of electrons with a line charge density $\lambda$ [charge per length] with a parallel speed $v$ such that the current is $I=\lambda v$. The magnetic field $B$ and the electric field $E$ a distance $r$ away from this infinite line current is
$$ B = \frac{\mu_0 I}{2\pi r}$$
$$ E = \frac{\lambda}{2\pi\epsilon_0 r}$$
As you state, the attractive magnetic force $f_B$ per length between the streams is
$$ f_B = B I = \frac{\mu_0 (\lambda v)^2}{2\pi r} $$
At the same time, the repellant electric force $f_E$ per length between the streams is
$$ f_E = E\lambda = \frac{\lambda^2}{2\pi\epsilon_0 r}$$
The total force per length is therefore (attractive positive, repellant negative)
$$ f_\mathrm{total}= f_B-f_E = -\frac{\lambda^2}{2\pi \epsilon_0 r^2}\Big(1-\frac{v^2}{c^2}\Big) $$
where $c=\frac{1}{\sqrt{\mu_0 \epsilon_0}}$ is the speed of light. In a relativistic calculation, one should replace $\lambda\rightarrow \frac{\lambda_0}{\sqrt{1-\frac{v^2}{c^2}}}$, due to the Lorentz contraction where $\lambda_0$ is the proper line charge density measured in the coordinate system where the electrons are at rest. The total force per length $l$ between the streams is therefore the same in all inertial frames,
$$ f_\mathrm{total} = -\frac{\lambda_0^2}{2\pi \epsilon_0 r^2} =- (f_E)_0/l$$
where $(f_E)_0/l$ is the electric force per length between the streams measured in the coordinate system where the electrons are at rest.
