Assuming only breadboard connection points which go straight down into the board (and at no other angle, such as connectors off to the side of the breadboard), there is a simple solution. Note that here I am talking only about the points needed to create the core of the wire path, not the renderable geometry itself.
First, consider the board top down and draw straight lines between the connection points. This step enables you to determine potential wire crossings. Add any unique crossed pairs to a list, for checking against one another, later.
Now, viewed side-on, you want to avoid crossings AND have a more wirelike (curved) profile, such that the connection curves from point A on the board, up, over and down to point B.
So what we do is to take that line and turn it into a semi-circle. (Formulae for getting points on a circle at various incremental radii can be found all over the internet.)
Great. Since most pairs of connections will not have the same length (in 2D), then even if these intersect on the top-down drawing, they usually (!) will not intersect side-on, because the different lengths lead to different radii which lead to different wire curvature.
If intersections of wires matter to you...
However, in some cases you may have an intersection. Though these will be rare (and rarer the thinner the final wire geometry is), if this does matter to you, you will need to check all cylindrical segments of each pair stored above, against one another, and determine if there are intersections - this could be costly, but it sounds like you are building your wires once-off. What you need to do here is scale the set of semi-circular vertices such that is is more elliptic. This perturbation needs to occur until they no longer intersect. To check whether two wires intersect, consider the diagram from top-down again: if the height of the two arcs is the same where the two 2D / overhead lines intersect, then you need to shift one or the other arc, accordingly.
Final cosmetic effects
Note that making these curves more elliptic may also be something you do just to get every wire looking a bit different, so it's not all too uniform. As a bonus step, you could also, instead of just using a semi-circle across the distance, use two rounded corners going down into the breadboard, and a straight section between.
Beyond this, with linear segments, perturbations etc. are yours to make as you wish, for example if you wanted a wire to look like it had been fiddled with by hand.
