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I don't know about the HP O scope, but if it uses a normal, electrostatically deflected, oscilloscope tube, the electrons speed will travel at nowhere near the speed of light.
The usual arrangement in the gun is that electrons leave the cathode relatively slowly and are attracted toward, and accelerated by, the anode and controlled by the grid in the same manner as a thermionic valve (tube). There are additional anodes which further accelerate and focus the beam.
In some 'scopes, there is a final anode after the deflection plates. This has the advantage that deflection is done when the electrons are moving relatively slowly and deflection is easier, then finally accelerated to max to hit the screen.