Updating observations of a signal due to the delays in the measurement
process is a common problem in signal processing, with prominent examples in a
wide range of fields. An important example of this problem is the nowcasting of
COVID-19 mortality: given a stream of reported counts of daily deaths, can we
correct for the delays in reporting to paint an accurate picture of the
present, with uncertainty? Without this correction, raw data will often mislead
by suggesting an improving situation. We present a flexible approach using a
latent Gaussian process that is capable of describing the changing
auto-correlation structure present in the reporting time-delay surface. This
approach also yields robust estimates of uncertainty for the estimated
nowcasted numbers of deaths. We test assumptions in model specification such as
the choice of kernel or hyper priors, and evaluate model performance on a
challenging real dataset from Brazil. Our experiments show that Gaussian
process nowcasting performs favourably against both comparable methods, and
against a small sample of expert human predictions. Our approach has
substantial practical utility in disease modelling -- by applying our approach
to COVID-19 mortality data from Brazil, where reporting delays are large, we
can make informative predictions on important epidemiological quantities such
as the current effective reproduction number.