Until recently (spring of 95) no one would have disagreed about the general thrust of BBN. There were occasional disagreements about data analysis, but it was generally understood that the field was one of the major successes of the standard big bang model. That may not be true anymore.
The focus of the disagreement seems to be the deuterium abundance.
If the model of Hata et al. 95 is correct, the implication is that the deuterium abundance is not in accord with the standard calculations. This leaves two important means of resolution, both of which require new physics. First, the effective number of relativistic degrees of freedom may be something other than two. This could come from heavy neutrinos, degeneracy, or a sterile neutrino family. Second, the model for calculation of nucleosynthesis may be wrong in some improtant manner. I know of no viable alternative currently in existance.
As an alternative to this, CST95 argues from their stochastic model that the primordial deuterium estimates of Hata et al. are flawed. In their initial analysis, the data are found to be concordant with BBN predictions. A more recent paper suggests that if the standard BBN model is correct, there is evidence for an increased systematic error in the measurments of the primordial helium 4 abundance.
The following graphs present the contention. First, from Hata et al. 95, their view of the problem. The shaded regions show the 65% confidence measurments in their analysis. The 95% regions are shown by the dotted loops.
Second, from CST95. Notice that the confidence reigons (in boxes in these graphs) are quite different, and allow for overlap.
As noted in other places, the ball is primarily in the court of the observationalists. A significant inprovement in the primordial abundance measurments could conclusively decide whether or not the crisis claimed by Hata et al. actually exists, and also provide strong constraints on any new physics. In short, BBN is a healthy science in the process of significant improvement.