Use a subgroup analysis in meta-analysis comparing studies before and after 2000, with a direct test of the between-subgroup effect difference. Alternatively use meta-regression with year, or the before/after indicator, as a study-level predictor. Compare the effect estimates themselves: significant and nonsignificant results need not differ significantly.
Two reasons for caution are that the proposed contrast was selected after examining the results, creating a multiple testing issue, and that only six studies supply an observational comparison of study characteristics. Year can be confounded with dose, participant populations, follow-up or design, so an apparent year effect need not be caused by time itself. The small number of studies also makes these comparisons imprecise. These would be exploratory findings rather than confirmation of an effect-modification hypothesis.