Answer
Speed does not change.
Work Step by Step
Let $q$ be the charge of the electron. Magnetic force on the electron = $q(\vec{v}\times \vec{B})$
Let in $dt$ time, infinitesimal displacement of electron is $\vec{v}dt$
Thus infinitesimal work done by magnetic field is $q(\vec{v}\times\vec{B})\cdot \vec{v}dt=q\vec{v}\cdot(\vec{v}\times \vec{B})dt=q(\vec{v}\times\vec{v})\cdot\vec{B}dt=0$
( as $\vec{v}\times \vec{v}=\vec{0}$ )
We use here the property of the scalar product of vectors.
As the infinitesimal work is zero, the overall work done by the magnetic field is zero.
By the work-energy theorem, the kinetic energy ( $K$ )of the electron cannot change in a magnetic field.
$K=\frac{1}{2}mv^2$ where $m$ is electron mass and $v$ is electron speed. As $K$ is unchanged and $m$ is a constant, that implies $v$ does change with time.
Thus, the speed cannot change with time.
