solar system - Do planets migrate suddenly or gradually?

Thommes et al. (2001) ran simulations and found that, at optimal conditions (namely, a planet of ~ 10 Earth masses), migration can be complete with ~ 100,000 years. Note that this was done before in-depth research was done on the Nice model, which is very similar. However, the mechanisms are different, as are the planet masses. The difference in timescales is dramatic.

Levison et al. (2007) did explore their own model - the Nice Model. They found that it took 60 million years to 1.1 billion years for Jupiter and Saturn to break their resonance. The period of encounters and scattering lasted for 878-885 million years, followed by a period of eccentricity damping lasting for 0.3 million years to 4 million years.

What was relatively quick was the ejection of the hypothetical 5th giant planet. The change in the other giants' orbits, however, was not.

So yes, planetary migration on this scale takes a long time. A very long time.

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For some really interesting results, see the graphs of semi-major axis vs. time from the various four-, five-, and six- planet models of Nesvorný & Morbidelli (2012). There are some incredible oscillation among the orbits of Uranus and Neptune in some of the simulations, which is eventually slowly damped.

Fig. 14.— Orbit histories of the giant planets in a simulation with five initial planets. See the caption of Fig. 1 for the description of orbital parameters shown here. The five planets were started in the (3:2,3:2,2:1,3:2) resonant chain, M_disk = 20 M_Earth and B(1). The fifth planet was ejected at $t =$ 6.1 Myr after the start of the simulation.

There's another relevant passage, if you're looking at setups with six giant planets:

The instability typically occurred in two steps, corresponding to the ejection of the two planets. Sometimes, as in Fig. 18, the ejection of the two planets was nearly simultaneous, but most of the times there was a significant delay between ejections. This was useful because the first planet’s ejection partially disrupted the planetesimal disk and reduced its capability to damp e55, which was then excited by the second planet’s ejection. While this mode of
instability can be important, we would need to increase the statistics (>100 simulations for each initial condition) to be able to properly resolve the small success fractions in the six-planet case.