I’m by no means an expert on the Wald distribution. But Google tells me it also likes to go by the name of Inverse Gaussian.
Wikipedia lists an algorithm for generating a random variate from an inverse Gaussian. It seems straight-forward enough to do. Except in your case you need some sort of shifting or censoring, so I’m not sure. Maybe you could generate the variate and reject those below the threshold or something.
Another way of doing it would be to write an inverse CDF function using the algebraic solver [going to go out on a limb here and claim there is no closed-form inverse CDF for this distribution] and then apply this cool trick by @lcomm.