It is not possible to determine the result of a collision between point particles in 2D or 3D. This is because the directions in which they rebound off each other depends on the orientation of the line or surface along which they make contact, relative to their previous directions of motion. However point particles have no structure so no such line or surface can be defined.
In 1D the point particles are constrained to move along a single line and can only rebound back along the direction from which they came.
This situation is familiar in billiards and snooker. A stationary ball can be made to move in a variety of directions depending on whether it is struck head on or to one side by the cue ball. For each ball the change in velocity is always directed perpendicular to the common surface at which they make contact.
The simplest solution for you would be to assign at random a line along which contact is made. Each particle retains its initial component of velocity parallel to this line. The final components of velocity perpendicular to this line are calculated from the laws of conservation of momentum and kinetic energy as is usual for 1D collisions. For elastic collisions (in which no kinetic energy is lost) the relative speed of separation (along the perpendicular) must equal the relative speed of approach, regardless of the relative masses.
A more sophisticated solution is to model the particles as circles with a small but finite radius. The common tangent along which these circles touch defines the line of contact.