The wavelength does of course remain the same. Think about two circles (representing wavefronts) evolving over time. Their respective radii increase at the same rate, such that the distance between them (the wavelength) always stays the same.
Now think about drawing a box of fixed size and placing it over the wavefronts. This represents your detector (or radio). At early times, with the box at a small distance from the centre, the wavefronts passing through the box might be noticeably curved. At later times when the wavefront radii are much larger, the same box would encapsulate what would approximate to two straight lines. In other words plane waves.
As perhaps you can see, the condition for this to be true is that the distance from the source must be much greater than the wavelength.
(Left) Expanding wavefronts at some time $t$. There is significant curvature seen inside the box (Right) Same wavefronts sometime later. Inside the box they are almost straight lines. NB: These spherically symmetric wavefronts would not be representative of an oscillating charge (which would have a dipole pattern), but whatever the wavefront shape, the argument about a decreasing radius of curvature holds true.