Despite the prevalence of transient-searching facilities operating across most wavelengths, the ultraviolet (UV) transient sky remains to be systematically studied. Therefore, we have 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/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 hours after the observations are performed), so we discover transients in near real-time. Transients which last >0.5 days are therefore still active when discovered, allowing for follow-up observations to be performed. From October 1, 2020, to the time of submission, using TUVOpipe we have processed 111 330 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. 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 so far obtained.