Ah yes, 'falling short' on places such as the New Cross Woolworths, leading to deaths of 168 persons, and laying waste areas such as Peckham where the fall of ten or more V1s and V2s were packed into a few streets - the area is still partly undeveloped. These unassuming residential and commercial areas took the shots intended for the centre, a few miles North.

The V2 had a guidance computer. Early rockets were purely inertial while some later rockets also rode radio beams to their targets.

The V2 used pendular integrating gyroscopic accelerometers (PIGAs) and gyroscopes for guidance. As far as I know, the V2 was the first use of the PIGA. Descendants of the V2 guidance systems were used by the Apollo moon missions and several US intercontinental ballistic missiles.

The PIGA is actually a pretty cool piece of machinery. You have a free-swinging balance arm that has a mass and a gyroscope on it. The gyroscope's axis of rotation is aligned with the balance arm. An electric motor spins the whole balance arm in a direction perpendicular to both the gyroscope's axis of rotation and the axis about which the balance is free to rotate. So if you remember from Freshman physics, you have a rotational momentum and an applied torque, the resultant torque is the cross product of the two (in other words, a torque in the direction the balance moves freely). An analog controller keeps the electric motor spinning at a rate that keeps the balance beam level. So, the rate the electric motor is turning is proportional to acceleration in the direction of the electric motor's axis. Thus, the integral of the acceleration is proportional to the integral of the rotational velocity of the electric motor. In other words, the number of revolutions of the electric motor is proportional to the velocity of the rocket (relative to free-fall). A simple revolution counter is sufficient to cut off the engine's fuel flow once the rocket has reached a given velocity.

(Source: I interned at Draper Lab, which made the Apollo guidance systems and several ICBM guidance systems. I walked by an Apollo PIGA in a display case every morning.)