We present source detection and catalogue construction pipelines to build the first catalogue of radio galaxies from the 270 $\rm deg^2$ pilot survey of the Evolutionary Map of the Universe (EMU-PS) conducted with the Australian Square Kilometre Array Pathfinder (ASKAP) telescope. The detection pipeline uses Gal-DINO computer vision networks (Gupta et al. 2024, PASA, 41, e001) to predict the categories of radio morphology and bounding boxes for radio sources, as well as their potential infrared host positions. The Gal-DINO network is trained and evaluated on approximately 5 000 visually inspected radio galaxies and their infrared hosts, encompassing both compact and extended radio morphologies. We find that the Intersection over Union (IoU) for the predicted and ground-truth bounding boxes is larger than 0.5 for 99% of the radio sources, and 98% of predicted host positions are within $3^{\prime \prime}$ of the ground-truth infrared host in the evaluation set. The catalogue construction pipeline uses the predictions of the trained network on the radio and infrared image cutouts based on the catalogue of radio components identified using the Selavy source finder algorithm. Confidence scores of the predictions are then used to prioritise Selavy components with higher scores and incorporate them first into the catalogue. This results in identifications for a total of 211 625 radio sources, with 201 211 classified as compact and unresolved. The remaining 10 414 are categorised as extended radio morphologies, including 582 FR-I, 5 602 FR-II, 1 494 FR-x (uncertain whether FR-I or FR-II), 2 375 R (single-peak resolved) radio galaxies, and 361 with peculiar and other rare morphologies. Each source in the catalogue includes a confidence score. We cross-match the radio sources in the catalogue with the infrared and optical catalogues, finding infrared cross-matches for 73% and photometric redshifts for 36% of the radio galaxies. The EMU-PS catalogue and the detection pipelines presented here will be used towards constructing catalogues for the main EMU survey covering the full southern sky.