Does the one-tailed vs two-tailed argument apply in GLMs (i.e. Wald test) in the same way it would in a Student’s t-test? I have constructed a gamma glm modelling hormone concentration with three predictors: treatment (treatment or control), sex (male or female) and site (thyroid, parathyroid). The work is mostly exploratory, and I had no a priori hypotheses regarding the effects of sex or site, but I have seen previously that the experimental treatment group produced higher hormone concentrations than the control, so I hypothesised a main effect of increased hormone concentrationin the experimental treatment group, over the control.
In my results I have reported the t statistic and p value for each predictor, but for treatment, my supervisor said this should be one-sided as I have an a priori hypothesis (that hormone concentration will be larger in treatment compared to control) and therefore my p value should be halved to turn it into a one-tailed value.
I know GLMs report a t statistic, but I believe it’s a Wald’s test it’s reporting which is slightly different from a t-test? Does the one-tailed vs two-tailed argument apply in GLMs in the same way it would in a Student’s t-test? Is it reasonable for me to simply halve my p-value in this case and report it as a one-tailed value?

(With a GLiM, strictly speaking, it's a $z$ statistic, not a $t$ statistic.)
The arguments about one-tailed vs. two-tailed work exactly the same in this context as the conventional STATS 101 $t$-test context.  For what it's worth, I typically default away from one-tailed tests.  In your case, I think you should have had a strong a-priori hypothesis, and have built that decision into the design of the study—it's sketchier to decide you want a one-tailed test after you've seen the results.
