They are 3 different algorithms: they work better in parallel, rather than in series because they have different purposes.
In addition to that, their output always brings some uncertainty (overall PCA), which will increase if reused in other algorithms.
PCA is mainly used to understand better the features: their variance and their linear correlations.
UMAP is non-linear and makes rational clusters for data exploration.
You have a navigator here to see the difference between PCA and UMAP.
Random Forest is an actual prediction or classification algorithm, but it depends on your data. In the case of time series, LSTMs or XGBoost could be better.
In conclusion, PCA and UMAP will grant you a better comprehension of your data that would allow you to make a good data preprocessing for your prediction algorithm.