Easy
In rectangular case 3x3 grid works for any aspect: if you have point (x,y) and look for the closest point in set (x_i+dx*m,y_i+dy*n), where (x_i,y_i) - given points and dx,dy - torus dimensions, in will be one of points (x_i+dx*round((x-x_i)/dx),y_i+dy*round((y-y_i)/dy). If (x,y) and (x_i,y_i) are in the same fundamental region, then |x-x_i|<dx, |y-y_i|<dy, so |round((x-x_i)/dx)|<=1. Actually you need only 2x2 grid
Non-rectangular torus case is more complicated: if shifts are (a,0) and (b,c), where |b|<a/2, b^2+c^2<a^2, take d=sqrt((a+|b|)^2+c^2)/2 and points from periodic pattern in the rectangular [min(0,b)-d,a+max(0,b)+d]*[-d,c+d]. Intersection of their Voronoi diagram with the fundamental region gives the answer.