Afneer, thank you for a nearly complete post.  Sorry I missed this the first time.

The solver makes good progress with many converged substeps up to a time of 0.6875 in this nonlinear buckling solution.

The only item missing from your post is the loading definition.

In a nonlinear with plasticity buckling solution that is loaded with force, it is quite normal for the solution to fail to converge at the point where the structure has reached the critical buckling load when the next load increment results in collapse. You just don't get to see that.   In models loaded with displacement, it is possible to continue solving into the post-collapse domain.  In models loaded with force, sometimes stabilization can achieve convergence in the post-collapse domain.

The clue that the solution has reached the point of collapse is that the force-displacement curve approaches a zero slope. Plot that curve and see if that is the case. If so, you have successfully identified the critical buckling load.  If there is a significant slope, then a numerical problem has stopped the convergence.