As I think you already know when using LinearModelFit the F-statistic and associated P-value show up but I don't know why that isn't included in the "ANOVATable" for NonlinearModelFit. However, the "ANOVATable" gets you the overall model information from which you can construct the F-statistic and associated P-value for the overall model:
data = {{0, 1}, {1, 0}, {3, 2}, {5, 4}, {6, 4}, {7, 5}};
nlm = NonlinearModelFit[data, Log[a + b x^2], {a, b}, x];
nlm["ANOVATable"]
anova = nlm["ANOVATableEntries"];
FStatistic = anova[[1, 3]]/anova[[2, 3]]
(* 45.1398 *)
Pvalue = 1 - CDF[FRatioDistribution[anova[[1, 1]], anova[[2, 1]]], FStatistic]
(* 0.00180005 *)
What is probably more interesting is to obtain the t-values and associated P-values for the individual parameters:
nlm["ParameterTable"]
