In infectious disease surveillance, incidence data are frequently subject to reporting delays and retrospective corrections, making it hard to assess current trends in real time. A variety of probabilistic nowcasting methods have been suggested to correct for the resulting biases. Building upon a recent comparison of eight of these methods in an application to COVID-19 hospitalization data from Germany, the objective of this paper is twofold. Firstly, we investigate how nowcasts from different models can be improved using statistical post-processing methods as employed, e.g., in weather forecasting. Secondly, we assess the potential of weighted ensemble nowcasts, i.e., weighted combinations of different probabilistic nowcasts. These are a natural extension of unweighted nowcast ensembles, which have previously been found to outperform most individual models. Both in post-processing and ensemble building, specific challenges arise from the fact that data are constantly revised, hindering the use of standard approaches. We find that post-processing can improve the individual performance of almost all considered models both in terms of evaluation scores and forecast interval coverage. Improving upon the performance of unweighted ensemble nowcasts via weighting schemes, on the other hand, poses a substantial challenge. Across an array of approaches, we find modest improvement in scores for some and decreased performance for most, with overall more favorable results for simple methods. In terms of forecast interval coverage, however, our methods lead to rather consistent improvements over the unweighted ensembles.