The OED asserts that the origin of radio as a stand-alone word arises by clipping earlier uses of radio as a combining form. The earliest stand-alone use that the OED cites is in 1903, whereas the earliest cited use as a combining form is this one:
1875 W. Crookes in Proc. Royal Soc. 23 377 The luminous rays..repel the black surface more energetically than they do the white surface. Taking advantage of this fact, the author has constructed an instrument which he calls a radiometer.
The title of that article is On attraction and repulsion resulting from radiation, and the terminology radiation occurs repeatedly throughout the article, including in the vicinity of the coined term radiometer. For example, this quote appears in the very next paragraph, which also contains the word rays:
The author finds that this instrument revolves under the influence of radiation, the rapidity of revolution being in proportion to the intensity of the incident rays.
Unless there's something even earlier, the best guess I could form on this basis is that the word "radio", rather than deriving directly from "radius", instead derives from Mr. Crookes determination that "radiometer" just sounded better than "ray meter" or "radiation meter".
Poking around in the future of 1875, the earliest entries in the OED for "radio" as a stand-alone word in the phrases "radio wave" and "radio frequency" appear at just about the same time, namely 1915, with their modern meaning. But the OED gives the following slightly earlier stand-alone use:
1912 Statutes U.S.A. XXXVII. i. 308 ‘Radio communication’ as used in this Act means any system of electrical communication by telegraphy or telephony without the aid of any wire [etc.].
and then, tracing the telegraph connection further back, there is this interesting entry:
1898 J. Munro in Electrician 21 Jan. 428/2 ‘Wireless telegraphy’ is not a bad technical term; but if a more scientific name be desirable would not Radiotelegraphy or Ray Telegraphy be preferable to ‘Space Telegraphy’? which Dr. Lodge employs.
This leads one to think that the "radio" combining form had become well established in science in the years since Crookes' radiometer. My best guess here might be that as the technology of telegraphic communication via electromagnetic waves advanced, the more general use of the combining form "radio-", introduced by Crookes' and referring to unspecified frequencies of radiation, somehow became narrowed down to refer to the familiar wavelengths of electromagnetic radiation in its modern meaning.