Despite the prevalence of transient-searching facilities operating across most wavelengths, the ultraviolet (UV) transient sky remains to be systematically studied. Therefore, we recently initiated the Transient Ultraviolet Objects (TUVO) project, with which we search for serendipitous UV transients in data obtained using currently available UV instruments with a strong focus on the UV and Optical (UVOT) telescope aboard the Neil Gehrels Swift Observatory (an overview of the project is described in a companion paper). Here, we describe the pipeline (named TUVOpipe) we constructed in order to find such transients in the UVOT data, using difference image analysis. The pipeline is run daily on all new public UVOT data (which are available 6–8 h after the observations are performed), so we discover transients in near real time. Transients that last >0.5 days are therefore still active when discovered, allowing for follow-up observations to be performed. From 01 October 2020 to the time of submission, we used the TUVOpipe to process 75 183 individual UVOT images, and we currently detect an average rate of ~100 transient candidates per day. Of these daily candidates, on average ~30% are real transients (separated by human vetting from the remaining “bogus” transients which were not discarded automatically within the pipeline). Most of the real transients correspond to known variable stars, though we also detect a significant number of known active galactic nuclei and accreting white dwarfs. The TUVOpipe can additionally run in archival mode, whereby all the archival UVOT data of a given field is scoured for ‘historical’ transients; in this mode, we also mostly find variable stars. However, some of the transients we find (in particular in the real-time mode) represent previously unreported new transients or undiscovered outbursts of previously known transients, predominantly outbursts from cataclysmic variables. In this paper, we describe the operation of (both modes of) TUVOpipe and some of the initial results we have obtained so far.