But they don't need the stars to physically collide for it to be a problem. A star coming anywhere near the orbit of Jupiter would pull planets away from the Sun, which would make dramatic changes to the Solar System.
If you reframe the question as "will the collision of the galaxies cause problems for some hypothetical civilizations who may be living there" the probability of that is simple, it's 1. The good news for such civilizations is that they'll have literally hundreds of thousands of years of warning to deal with them. Planets may have problems but if a civilization is based on space stations and other off-planet structures they'll hardly notice the problems since they'll be so slow to occur.
Correct me if I’m wrong, but isn’t the scale of our solar system minuscule compared to the distance between stars? Presumably if another star gets close enough to affect Jupiter, it’s going to affect everything to varying degrees.
Yes you're correct! It's about a factor of 50,000 between the orbit of Jupiter and the distance to the nearest star.
But the orbit of Jupiter is still a lot bigger than the size of the Sun. To give numbers: The radius of the Sun is 700km, the radius of the orbit of Jupiter is 7*10^8 km (approx, it varies a bit), and the distance to the nearest star is 4*10^13 km.
