Defining a nautical mile Astronomy is new to me so my question might be stupid: I read that a nautical mile is defined as one minute of latitude along any line of longitude. What if it was the other way round, i.e. one minute of longitude along any line of latitude? Would it make any difference?
Yes, very much so.
The lines or circle of constant longitude always form a great arc which intersects with the poles; thus it always has the same length. A circle of constant latitude varies in circumference: the largest is at the equator while it has intermediate length at intermediate latitudes and no length anymore when you reach the pole.

See e.g. that image from wikipedia and follow a circle in longitude, thus from North to South. And compare the different sizes of different circles for different lattitudes (thus the circles which are parallel to the equator)
Mind though that the nautical mile is no longer defined via the fraction of the polar circumference of Earth. It is defined as exactly 1852m in length, and thus can in principle be determined as accurately as one can measure time (1852 / 299 792 458 seconds).
In earlier times it was a MUCH easier task to determine a difference in latitude as it directly relates to the (culmination) height of celestial objects above the horizon. It was VERY DIFFICULT to impossible to determine difference in longitude unless you had a very accurate clock and an almanach of the rise and set times of celestial objects.