I cannot give you one expression, but here are several articles that cover some non-normal cases:
Browne, M. W., & Shapiro, A. (1986). The asymptotic covariance matrix of sample correlation coefficients under general conditions. Linear Algebra and its Applications, 82, 169-176.
Gayen, A. K. (1951). The frequency distribution of the product-moment correlation coefficient in random samples of any size drawn from non-normal universes. Biometrika, 38, 219-247.
Kowalski, C. (1972). On the effects of non-normality on the distribution of the sample product-moment correlation coefficient. Applied Statistics, 21, 1-12.
Subrahmaniam, K., & Gajjar, A. V. (1980). Robustness to nonnormality of some transformations of the sample correlation coefficient. Journal of Multivariate Analysis, 10, 60-77.
Yuan, K.-H., & Bentler, P. M. (2000). Inferences on correlation coefficients in some classes of nonnormal distributions. Journal of Multivariate Analysis, 72, 230-248.
Yes. Instead of using a two-sided critical value from a t-distribution with $n-2$ degrees of freedom (e.g., $\pm 2.09$ for $n=22$ and $\alpha=.05$, two-sided), you would use just the upper critical value (e.g., $+1.72$ for $n=22$ and $\alpha=.05$, one-sided).
Best Answer
The statistic you've mentioned $t=\frac{r}{\sqrt{(1-r^2)/(N-2)}}$ is essentially a significance of regression testing problem under the given setting :-
Let $(x_i, y_i)$ be a bivariate normal random sample of size $N$. Then we know that $E(Y|X=x)=\left(\mu_Y-\rho \dfrac{\sigma_Y}{\sigma_X} \mu_X\right)+\left(\rho \dfrac{\sigma_Y}{\sigma_X}\right)x = \beta_0 + \beta_1x$
Where $\beta_0 = \left(\mu_Y-\rho \dfrac{\sigma_Y}{\sigma_X} \mu_X\right)$ and $\beta_1 = \left(\rho \dfrac{\sigma_Y}{\sigma_X}\right)$
Now to test $H_0 : \rho = 0$ is equivalent to testing $H_0 : \beta_1 = 0$.
Let $\hat{\beta_1}$ be the OLS estimator of $\beta_1$ then under $H_0 : \beta_1 =0$
$t=\frac{\hat{\beta_1}}{s.e(\hat{\beta_1})}=\frac{r}{\sqrt{(1-r^2)/(N-2)}}$ follows a t-distribution with $N-2$ degrees of freedom.
Hence the only thing you need to ensure is that the samples are independent and follow Bivariate Normal Distribution. Note that checking for individual normality will not work as marginal normality doesn't imply joint normality.