We present extensive, high-density Swift observations of V2491 Cyg (Nova Cyg 2008 No. 2). Observing the X-ray emission from only one day after the nova discovery, the source is followed through the initial brightening, the Super-Soft Source phase and back to the pre-outburst flux level. The evolution of the spectrum throughout the outburst is demonstrated. The UV and X-ray light-curves follow very different paths, although changes occur in them around the same times, indicating a link between the bands. Flickering in the late-time X-ray data indicates the resumption of accretion. We show that if the white dwarf is magnetic, it would be among the most magnetic known; the lack of a periodic signal in our later data argues against a magnetic white dwarf, however. We also discuss the possibility that V2491 Cyg is a recurrent nova, providing recurrence timescale estimates.