Early detection of the SARS-CoV-2 virus in human mucus is now an urgent need to settle down the COVID-19 outbreaks. These global pandemic outbreaks are hard to stop without massive testing. One promising solution is non-intrusive collection of data in the form of cough sounds and using various speech processing techniques for classifying diseased samples from healthy ones. In this paper we present a non-invasive, quick and efficient pre-screening methodology using cough sounds for the detection of COVID-19, employing cepstral domain analysis viz. Mel-Frequency Cepstral Coefficients (MFCC), Gammatone Frequency Cepstral Coefficients (GFCC) and Human Factor Cepstral Coefficients (HFCC) using architectures such as 1-D Convolutional Neural Network (CNN), Recurrent Neural Network (RNN), Long Short-Term Memory (LSTM) and Bidirectional Long Short-Term Memory (BiLSTM). Subsequently, we analyse the impact of feature fusion technique for the classification of cough audio data as COVID-19 affected or not affected categories and we have found that the model with feature selection using Spearman Rank Correlation on combined cepstral features gave highest accuracy 72% with LSTM.