Hope it was helpful.

There is a different interpretation between the $R^2$ calculated using LinearModelFit and NonlinearModel but not because of something intrinsically different between linear and nonlinear models but rather that Mathematica chooses to use two different formulas for $R^2$ for the two procedures.

LinearModelFit uses $1-{{SS_{res}}\over{SS_{corrected Total}}}$ and NonlinearModelFit uses $1-{{SS_{res}}\over{SS_{uncorrected Total}}}$. One can see this by fitting the same linear model in LinearModelFit and NonlinearModelFit.

Thank you very much. This is very helpful. The data I am working with is the percentage of "anocracies" (between democracy and autocracy) in the Polity IV database of regime types from 1800 to 2017, which you would not expect to be sinusoidal. It looks like it would probably fit a saw tooth wave as well as or better than a sine wave. However, a sine wave is the solution to a simple dynamic model (second derivative proportional to negative of current value), and so its presence provides a good starting point for developing a theory of what is going on.

The issue about R2 having a different interpretation is in the Mathematica Tutorial on Statistical Model Analysis. It says, "The coefficient of determination does not have the same interpretation as the percentage of explained variation in nonlinear models as it does in linear models because the sum of squares for the model and for the residuals do not necessarily sum to the total sum of squares." I'm not sure why this is, but I suppose it's to do with the non-linearity. It would be nice if they said how we might interpret it, but they don't...perhaps because it depends entirely on the situation.

What question do you want associated with a test of significance? Is it about specific parameters? Linear combinations of parameters? Predictions? NonlinearModelFit can certainly perform the appropriate test or provide information to be able to construct an appropriate test. One just needs to be specific about what you want to test and under what conditions.

Tests of significance aren't necessarily very useful unless you have some idea as to what kind of difference from a hypothesized value you're looking for. Maybe estimation rather than hypothesis testing is what might be more appropriate for your needs.

Also, "the probability of getting such a fit by chance is p" is not quite right about resulting P-values. A P-value is the probability of observing a test statistic at least as extreme as what you observed when a specified null hypothesis about the value of some unknown quantity is true.