As the question states, how would you calculate the t-statistics in GLS? Is it just the same as in OLS? And are these results still reliable?
I have heteroskedastic data and so am using GLS, but I'm not sure if the calculations for t-statistics change or the values are less reliable?
The t-statistics are computed in the usual manner: take $\hat\beta_i$ and divide it by the square root of the ith diagonal of the estimated GLS covariance matrix.
By the way, in a linear regression setting you do not have to perform GLS. Modern approaches favor OLS with standard errors adjusted to reflect the heteroskedasticity, as available in any reliable statistical package.
OLS will go through the means of the data, which seems most sensible. GLS will not.