I have a $6 (shipped) wired computer that I bought off eBay on my road bike. It works fine. I think if you're going to go wireless, you need to buy a really good computer or it's going to be badly susceptible to interference. At the $20 level it's probably just sending plain old CW pulses on a frequency, not keyed digital signals or anything of the sort, so any RF noise is going to just fritz it right out.
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Work: the 8 hours that separates bike rides.