Let's say I'm the student suspected of cheating and look at this from my perspective:
Case 1: I cheated, and receive this email warning me I came close to being caught. This should be a good result. It would act as deterrence - after all, next time I might not be so lucky. Only thing to pay attention to here is to not reveal the evidence against me, since otherwise I might be tempted to cheat again except this time also to cover any tracks.
Case 2: I didn't cheat. In this case the warning is irrelevant to me. Like, the conditional is false, so whatever follows it is irrelevant. After all, the statement "if you cheated then pigs fly" is true! The allegation would only become a nuisance if I had to invest time and effort to fend it off (which I presumably don't in this case), or if it became public (which it doesn't have to, if you send private email).
If you are very worried about case 2 (which is reasonable), you could try looking around to see the effect of false allegations in online forums such as StackExchange. We do for example have rules against multiple accounts. One can never actually be sure that two accounts are actually the same person, which will invariably lead to false accusations and possibly investigations. I don't actually have the statistics on how people react, but perhaps a mod will be able to say. My experience elsewhere is that people get annoyed if they have to mount a defense, but if it's just being told they've been suspected, they might even react with laughter.
Edit: Based on the comments, different people come to different conclusions on this because they interpret things differently. I for example would consider the proposed email closer to being exonerated than being accused, but many others don't. Clearly the dividing line between the two very different reactions is extremely fine. If you do send the letter, you will have to word it very carefully. At that point, it might be better to just sidestep the potential quagmire entirely and not bother.