The transverse velocity (proper motion) of M31 is unknown, but can be constrained, as demonstrated here:
Constraints on the Proper Motion of the Andromeda Galaxy Based on the Survival of Its Satellite M33
Abraham Loeb, et al., Astrophysical Journal 633(2): 894-898, Nov 2005
Abstract: A major uncertainty in the dynamical history of the Local Group of galaxies originates from the unknown transverse speed of the Andromeda galaxy (M31) relative to the Milky Way. We show that the recent VLBA measurement of the proper motion of Andromeda's satellite, M33, severely constrains the possible values of M31's proper motion. The condition that M33's stellar disk has not been tidally disrupted by either M31 or the Milky Way over the past 10 billion years favors a proper motion amplitude of 100+/-20 km s-1 for M31 with the quadrant of a negative velocity component along right ascension and a positive component along declination strongly ruled out. This inference can be tested by future astrometric measurements with SIM, Gaia, or the SKA. Our results imply that the dark halos of Andromeda and the Milky Way will pass through each other within the next 5-10 billion years.
I don't know how many orbits that amounts to, or if it even amounts to more than one. But 5-10 billion leaves a lot of wiggle room. In 5 billion the sun should be a red giant, or very close to it. So if the collision happens on the early side, that could be a real show. But if it takes 10 billion, then the sun will be a white dwarf, and maybe it will all be a tad less spectacular around here.
I also don't know of any such fine scale simulations of galaxy collisions. Usually these billion-body-problems are necesarily restricted to the gravitational interactions of the stars, or collisions between the large clouds of dust & gas (
Iono, Yun & Ho, 2005;
Iono, Yun & Mihos, 2004;
Braine, et al., 2004;
de Gris, 2003).
Colliding spiral galaxies NGC6621 & NGC 6622 (
Arp 81), or maybe the
Antennae Galaxies, are probably as good an example of any as to how a collision between M31 and the Milky Way will look. There is a lot of vigorous star formation where the clouds collide. But the stars rarely collide, and often produce tidal streams like the one in the Antennae. I think it is reasonable that in our local neighborhood, the extreme tidal disturbances could easily collapse large portions of the Oort cloud, and give us a rain of comets. There are several papers already on the books, showing that galactic tides are enough to trigger Oort Cloud comets to fall into the inner solar system, especially as the solar system passes through denser arms of the Galaxy (i.e.,
Napier, 2006;
Dybczynski, 2005;
Matese & Lissauer, 2004;
Dybczynski, 2002).