In the face of the global pandemic caused by the disease COVID-19, researchers have increasingly turned to simple measures to detect and monitor the presence of the disease in individuals at home. We sought to determine if measures of neuromotor coordination, derived from acoustic time series, as well as phoneme-based and standard acoustic features extracted from recordings of simple speech tasks could aid in detecting the presence of COVID-19. We further hypothesized that these features would aid in characterizing the effect of COVID-19 on speech production systems. A protocol, consisting of a variety of speech tasks, was administered to 12 individuals with COVID-19 and 15 individuals with other viral infections at University Hospital Galway. From these recordings, we extracted a set of acoustic time series representative of speech production subsystems, as well as their univariate statistics. The time series were further utilized to derive correlation-based features, a proxy for speech production motor coordination. We additionally extracted phoneme-based features. These features were used to create machine learning models to distinguish between the COVID-19 positive and other viral infection groups, with respiratory- and laryngeal-based features resulting in the highest performance. Coordination-based features derived from harmonic-to-noise ratio time series from read speech discriminated between the two groups with an area under the ROC curve (AUC) of 0.94. A longitudinal case study of two subjects, one from each group, revealed differences in laryngeal based acoustic features, consistent with observed physiological differences between the two groups. The results from this analysis highlight the promise of using nonintrusive sensing through simple speech recordings for early warning and tracking of COVID-19.