A nano abnormality detection scheme (NADS) in molecular nano-networks is studied. This is motivated by the fact that early detection of diseases such as cancer play a crucial role in their successful treatment. The proposed NADS is in fact a two-tier network of sensor nano-machines (SNMs) in the first tier and a data-gathering node (DGN) at the sink. The SNMs detect the presence of competitor cells (abnormality) by variations in input and/or parameters of a nano-communications channel (NCC). The noise of SNMs as their nature suggest is considered correlated in time and space and herein assumed additive Gaussian. In the second step, the SNMs transmit micro-scale messages over a noisy micro communications channel (MCC) to the DGN, where a decision is made upon fusing the received signals. We find an optimum design of detectors for each of the NADS tiers based on the end-to-end NADS performance. The detection performance of each SNM is analyzed by setting up a generalized likelihood ratio test. Next, taking into account the effect of the MCC, the overall performance of the NADS is analyzed in terms of probabilities of misdetection and false alarm. In addition, computationally efficient expressions to quantify the NADS performance is derived by providing respectively an approximation and an upper bound for the probabilities of misdetection and false alarm. This in turn enables formulating a design problem, where the optimized concentration of SNMs in a sample is obtained for a high probability of detection and a limited probability of false alarm. The results indicate that otherwise ignoring the spatial and temporal correlation of SNM noise in the analysis, leads to an NADS that noticeably underperforms in operations.