It all depends on the closest approach of any stars to the Sun.
When galaxies collide it is not that their stars crash into each other, because their individual cross-sections are extremely small when compared to the space between them. This is dealt with in qualitative terms on the wikipedia page on the likely collision.
The Milky Way disk at the position of the Sun has of order $n=0.1$ stars $pc^{-3}$ and is about $L=200$ pc thick. If we say the cross-section for something catastrophic to happen is a disk of radius 100 au around the Sun (at closer than this separation, another star would start to disrupt the quasi-equilibrium of the solar system planets) - this gives $\sigma \simeq 10^{-6}$ pc$^2$. If we then assume that the Andromeda disk is similar to ours (in terms of stellar density) and they have a disk-to-disk collision, then the "optical depth" for a collison is $n \sigma L \simeq 2\times 10^{-5}$, which is roughly the (small) chance of direct collision.
On the other hand, a significant perturbation to the Oort cloud might occur if a star came within say 0.1 pc, in which case, the optical depth for this is 0.6. So actually, it seems to me that a significant Oort cloud perturbation is rather likely, which could result in a very significant increase in the bombardment of the inner solar system by comets. I don't think the speed of the encounter matters too much: (i) The Oort cloud is supposed to be spherically symmetric; (ii) even for a relative velocity difference of 10 km/s, the encounter would take $<10^{4}$ years, so the target is effectively stationary.
Another potential consequence is a sudden increase in the star formation rate in the solar vicinity. "Starburst galaxies" (e.g. see the Antennae) can result from the merger of gas-rich spiral galaxies. The Milky Way and Andromeda partially fit that description. The caveat being that much of their gas will have been consumed in about 4 billion years. Nevertheless there remains a chance (which I would not know where to begin the calculation), that a burst of nearby star formation could result in the formation of massive stars with their consequent supernovae explosions. This might have a profound effect on any life still existing in the solar system.