Does the motor speed change with the fully automatic star(wye)/delta starter?
Speed will increase but not dramatically. A 3ph induction motor never runs synchronous with the AC power frequency; there is always the phenomena known as "slip" and slip is needed to induce a voltage into the rotor that in turn circulates a current and generates a rotor magnetic field. 

The amount of slip is dependant on mechanical loading of the rotor; more loading means more slip and a reduction in rotor speed and, even on no mechancial load, there is friction and windage losses just due to the rotor spinning.

If you then re-connect from star to delta formation you are providing more torque and that can overcome a part of these "no load" mechanical losses and hence the speed increases a little bit.

The picture below shows the effect of three different voltages and the resultant change in speed (given a particular torque-speed load): -