Fastest known complexity for combinatorial ILP algorithm?
From what I can tell by searching, the definite survey seems to be:

Aardal, Karen, Robert Weismantel, and Laurence A. Wolsey. "Non-standard approaches to integer programming." Discrete Applied Mathematics 123.1 (2002): 5-74.

In particular, Section 2.1 discusses integer programming in bounded dimension, and presents algorithms due to different authors. Indeed, the survey lists many references, and discusses some practical implementations.

For a fixed number of variables, integer linear programming is polynomial time solvable by Lenstra's algorithm.