For the past couple of decades, the Parkes radio telescope has been regularly observing the millisecond pulsars in 47 Tucanae (47 Tuc). This long-term timing programme was designed to address a wide range of scientific issues related to these pulsars and the globular cluster where they are located. In this paper, the first of a series, we address one of these objectives: the characterization of four previously known binary pulsars for which no precise orbital parameters were known, namely 47 Tuc P, V, W and X (pulsars 47 Tuc R and Y are discussed elsewhere). We determined the previously unknown orbital parameters of 47 Tuc V and X and greatly improved those of 47 Tuc P and W. For pulsars W and X we obtained, for the first time, full coherent timing solutions across the whole data span, which allowed a much more detailed characterization of these systems. 47 Tuc W, a well-known tight eclipsing binary pulsar, exhibits a large orbital period variability, as expected for a system of its class. 47 Tuc X turns out to be in a wide, extremely circular, 10.9-d long binary orbit and its position is ~3.8 arcmin away from the cluster centre, more than three times the distance of any other pulsar in 47 Tuc. These characteristics make 47 Tuc X a very different object with respect to the other pulsars of the cluster.