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For a disk of radius a, mass m and tilt alpha from the table, steady rolling with the centre of mass at rest requires the angular velocity to lie along the line from the contact point to the centre. Balancing the gravity-normal torque against the rate of change of angular momentum (inertia about a diameter m a^2/4) gives the precession rate
Omega^2 = 4 g / (a sin alpha)
The total energy is E = (3/2) m g a sin alpha. Any dissipation lowers E, so alpha falls, which in turn raises Omega: the wobble accelerates while the disk flattens. With loss power P = mu m g a Omega cos alpha (rolling resistance) the tilt obeys d alpha/dt = -(2 mu/3) Omega; with P = k m g a sqrt(a/g) Omega^2 cos alpha it obeys d alpha/dt = -(8k/3) sqrt(g/a) / sin alpha, which integrates in closed form: t0 = 3 sqrt(a/g) (1 - cos alpha0) / (8k).
Simplifications: the model assumes quasi-steady rolling without slipping on a flat table, a rigid thin disk and a phenomenological loss law. Real coins end with a buzzing, slipping and bouncing stage (and air-film damping) that this model does not capture, so the singularity here is an idealised limit, stopped at 0.5 degrees. The rate exponents depend on the loss law you choose; they are not a fit to a measured coin.