Spectrum sensing constitutes a key ingredient in many cognitive radio paradigms in order to detect and protect primary transmissions. Most sensing schemes in the literature assume a time-invariant channel. However, when operating in low Signal-to-Noise Ratio (SNR) conditions, observation times are necessarily long and may become larger than the coherence time of the channel. In this paper the problem of detecting an unknown constant-magnitude waveform in frequency-flat time-varying channels with noise background of unknown variance is considered. The channel is modeled using a basis expansion model (BEM) with random coefficients. Adopting a generalized likelihood ratio (GLR) approach in order to deal with nuisance parameters, a non-convex optimization problem results. We discuss different possibilities to circumvent this problem, including several low complexity approximations to the GLR test as well as an efficient fixed-point iterative method to obtain the true GLR statistic. The approximations exhibit a performance ceiling in terms of probability of detection as the SNR increases, whereas the true GLR test does not. Thus, the proposed fixed-point iteration constitutes the preferred choice in applications requiring a high probability of detection.