This article introduces the class of continuous time locally stationary wavelet processes. Continuous time models enable us to properly provide scale-based time series models for irregularly-spaced observations for the first time, while also permitting a spectral representation of the process over a continuous range of scales. We derive results for both the theoretical setting, where we assume access to the entire process sample path, and a more practical one, which develops methods for estimating the quantities of interest from sampled time series. The latter estimates are accurately computable in reasonable time by solving the relevant linear integral equation using the iterative soft-thresholding algorithm due to Daubechies, Defrise and De~Mol. Appropriate smoothing techniques are also developed and applied in this new setting. Comparisons to previous methods are conducted on the heart rate time series of a sleeping infant. Additionally, we exemplify our new methods by computing spectral and autocovariance estimates on irregularly-spaced heart rate data obtained from a recent sleep-state study.