You are confusing 2 separate types of oscillation. What I described in my last post was not a change in tilt angle. Imagine a line from the South Pole through to the North Pole, then projecting out into space. It makes an angle of 66 and a half degrees to the plane of the Earth's orbit round the Sun (or 23 and a half degrees to the vertial from the plane of the Earth's orbit round the Sun).
Like the child's toy gyro, that imaginary line rotates about the vertical to the plane of the Earth's orbit round the Sun. At present, it's pointing fairly closely towards Polaris (within about 1 degree). It will be at its closest to the direction of Polaris in 2017. however, at other points in the 26000 year cycle, it points to other stars - in fact sailors in classical Greek times used Vega as the Pole Star.
This type of precession, however, is separate from changes in the angle that the Earth's spin axis makes with the plane of its orbit round the Sun. This is a period of 41,000 years and the angle varies from 21.8 degrees to 24.4 degrees. At present the angle is about 23.4 degrees.