The apical four-chamber (A4C) view in fetal echocardiography is a prenatal examination widely used for the early diagnosis of congenital heart disease (CHD). Accurate segmentation of A4C key anatomical structures is the basis for automatic measurement of growth parameters and necessary disease diagnosis. However, due to the ultrasound imaging arising from artefacts and scattered noise, the variability of anatomical structures in different gestational weeks, and the discontinuity of anatomical structure boundaries, accurately segmenting the fetal heart organ in the A4C view is a very challenging task. To this end, we propose to combine an explicit Feature Pyramid Network (FPN), MobileNet and UNet, i.e., MobileUNet-FPN, for the segmentation of 13 key heart structures. To our knowledge, this is the first AI-based method that can segment so many anatomical structures in fetal A4C view. We split the MobileNet backbone network into four stages and use the features of these four phases as the encoder and the upsampling operation as the decoder. We build an explicit FPN network to enhance multi-scale semantic information and ultimately generate segmentation masks of key anatomical structures. In addition, we design a multi-level edge computing system and deploy the distributed edge nodes in different hospitals and city servers, respectively. Then, we train the MobileUNet-FPN model in parallel at each edge node to effectively reduce the network communication overhead. Extensive experiments are conducted and the results show the superior performance of the proposed model on the fetal A4C and femoral-length images.