We present an experimental investigation into the effectiveness of transfer learning and bottleneck feature extraction in detecting COVID-19 from audio recordings of cough, breath and speech. This type of screening is non-contact, does not require specialist medical expertise or laboratory facilities and can be deployed on inexpensive consumer hardware. We use datasets that contain recordings of coughing, sneezing, speech and other noises, but do not contain COVID-19 labels, to pre-train three deep neural networks: a CNN, an LSTM and a Resnet50. These pre-trained networks are subsequently either fine-tuned using smaller datasets of coughing with COVID-19 labels in the process of transfer learning, or are used as bottleneck feature extractors. Results show that a Resnet50 classifier trained by this transfer learning process delivers optimal or near-optimal performance across all datasets achieving areas under the receiver operating characteristic (ROC AUC) of 0.98, 0.94 and 0.92 respectively for all three sound classes (coughs, breaths and speech). This indicates that coughs carry the strongest COVID-19 signature, followed by breath and speech. Our results also show that applying transfer learning and extracting bottleneck features using the larger datasets without COVID-19 labels led not only to improve performance, but also to minimise the standard deviation of the classifier AUCs among the outer folds of the leave-$p$-out cross-validation, indicating better generalisation. We conclude that deep transfer learning and bottleneck feature extraction can improve COVID-19 cough, breath and speech audio classification, yielding automatic classifiers with higher accuracy.