TL/DR: Over 2000 years for a ridiculously advanced craft travelling at the surface escape velocity of the sun, 620 km/s.
The effectiveness of a solar sail is determined by its "lightness factor", the thrust-to-weight ratio under solar gravity. (Since both gravity and radiation pressure obey inverse square law, this is independent of distance from the sun.) It is sometimes given in terms of the characteristic acceleration or characteristic thrust, which is the value at Earth's distance from the sun. The characteristic acceleration due to solar gravity is 0.005 m/s^2. (Edit: I should have written 0.0059 as 0.006. The basic conclusion will remain the same.)
The speed that can be achieved is limited by a number of factors. The maximum temperature that the sail can withstand (which limits how closely it can fly by the sun to gain speed) is one limit. Material strength is also an important factor, such a very thin material could easily be torn apart by radiation pressure when pulling against the inertia of a spacecraft near the surface of the sun. Furthermore, dust so near the sun is not that well studied and may pose a risk.
As an example, Wolfgang Seboldt & Bernd Dachwald discuss a number of solar sails in "Solar Sails: Propellantless Propulsion for Near- and Medium-Term Deep-Space Missions" (*1) One currently build-able of 400 m^2 at 87 g/m^2, one of 2500 m^2 at 38 g/m^2 likely build-able if existing technology is refined and one of 4900 m^2 at 23 g/m^2 that would require significant research but is considered realistically achievable. (Payload mass not included.) The third could achieve a characteristic thrust of about 40mN, giving a lightness factor of β = a/(0.005 m/s^2 ) = 0.07.
This is not enough to reach Proxima in any reasonable amount of time unless the sail is accelerated with large lasers. Instead of working it all out, I will show what happens with a lightness factor of 1 (*2), much higher than the craft above even without payload. If the spacecraft is on an orbit around the sun (ideally highly eccentric) and deploys its solar sail approximately at perihelion, we may assume that the sail exactly cancels solar gravity for the entire trip. This means that the spacecraft will escape the solar system with a constant speed equal to whatever its speed at perihelion was. Unless we also use another propulsion system, we must assume that we start from a bound orbit and so this can at most be the surface escape velocity of the sun, 617 542 m/s (*3).
(3.991 * 10^16 m)/(6.18 * 10^5 m/s) = 6.46 * 10^10 s, or about 2000 years.
The acceleration period for such a β = 1 craft will consist of the period where it falls in towards the sun, after which it will as already mentioned escape at constant speed. Falling in from Earth would give much less than the mentioned speed; if we assume a Jupiter flyby was used for perihelion lowering then the acceleration time (starting at Jupiter) would be just about a sixth of a Jupiter year, or two years. (Edit: Egg on my face. I wrote half a Jupiter year, when I should have said half an orbit at half of Jupiter's semi-major axis. That makes for 0.5 * 0.5^(3/2) of Jupiter's orbital period.)
(*1) Published in "Advanced Propulsion Systems and Technologies Today to 2020" edited by Claudio Bruno & Antonio Accettura, 2008. (American Institute of Aeronautics and Astronautics) p. 448-449
(*2) See "The Startflight Handbook: A Pioneer's Guide to Interstellar Travel" by Eugene Mallove & Gregory Matloff, 1989. (Wiley) p. 96-97.
(*3) From "Space Mission Engineering: The New SMAD" edited by James Wertz et al., 2011. (Microcosm) p. 955
