We present a detailed study of the star formation history (SFH) of the Tucana dwarf spheroidal galaxy. High-quality, deep HST/ACS data, collected in the framework of the LCID project, allowed us to obtain the deepest color-magnitude diagram to date, reaching the old main-sequence turnoff (F814 ~ 29) with good photometric accuracy. Our analysis, based on three different SFH codes, shows that Tucana is an old and metal-poor stellar system, which experienced a strong initial burst of star formation at a very early epoch (sime13 Gyr ago) which lasted a maximum of 1 Gyr (sigma value). We are not able to unambiguously answer the question of whether most star formation in Tucana occurred before or after the end of the reionization era, and we analyze alternative scenarios that may explain the transformation of Tucana from a gas-rich galaxy into a dSph. Current measurements of its radial velocity do not preclude that Tucana may have crossed the inner regions of the Local Group (LG) once, and so gas stripping by ram pressure and tides due to a close interaction cannot be ruled out. A single pericenter passage would generate insufficient tidal heating to turn an originally disky dwarf into a true dSph; however, this possibility would be consistent with the observed residual rotation in Tucana. On the other hand, the high star formation rate measured at early times may have injected enough energy into the interstellar medium to blow out a significant fraction of the initial gas content. Gas that is heated but not blown out would also be more easily stripped via ram pressure. We compare the SFH inferred for Tucana with that of Cetus, the other isolated LG dSph galaxy in the LCID sample. We show that the formation time of the bulk of star formation in Cetus is clearly delayed with respect to that of Tucana. This reinforces the conclusion of Monelli et al. that Cetus formed the vast majority of its stars after the end of the reionization era implying, therefore, that small dwarf galaxies are not necessarily strongly affected by reionization, in agreement with many state-of-the-art cosmological models. Based on observations made with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. These observations are associated with program 10505.