The idea of a "spelling alphabet" of course assumes the concepts of "spelling" and "alphabet", which does not apply to Chinese characters.
That is not to say that radiotelephony procedure is unimportant. The need for a standard English in international aviation is so important that there is much research in China into teaching ICAO English standards, and their standards are to be adhered to in aviation environments.
Instead of spelling words, Chinese speakers in general explain the characters in terms of other words, a kind of 解说. Chinese speakers do quite a lot of this when encountering names, much like many English speakers ask the spelling of the name (or have various shortcuts). These are very far from standard in the way that NATO or ICAO are, and there are plenty of ad hoc cultural variants; but generally they use high-frequency disyllabic words. If the character is rare enough not to have such words, then other ad hoc explanations are required.
One must then go back to the reason for a spelling alphabet in alphabet- and syllabary-using language systems: when the technology of audio transfer causes the risk of information loss to be large enough to impede communication. It is a form of disambiguation.
This was true in late 1890s Britain, which resulted in the 1904 British Army Regulations, is still true in aviation contexts, and is also true of the Sinosphere. But since the information being lost is encoded slightly differently in the written script, the actions developed to compensate are different.
The disambiguation aspect is particularly obvious with numbers in Mandarin Chinese. The exact same vowel and tone shared by the numbers 1 and 7 mean that they are very easily confused over a poor line, so much so that in civilian conversation when enunciating numbers, 幺 is the usual substitution for 一 when reading phone numbers and other long strings of numbers (as well as other specific examples). The Chinese Wikipedia gives a few more examples of the numbers substituted in technical contexts.