Complexity of nearest codeword in cyclic codes
The general Nearest Codeword Problem problem for an arbitrary code is NP-hard. It is known that the problem is in NP if you only need to check whether there is a codeword within some fixed distance $\delta.$ 

Moreover there some efficient approximation algorithms given by Alon, Panigrahy and Yekhanin available here.

I am unaware of any work using the cyclicity property. The above apply to arbitrary linear codes.