I think one has to decide if time to reaction to drug A is censored or if a reaction to drug A just never happens for some subjects. If the latter is the case, you could use a linear predictor as the one James suggests with some functional form of the time to reaction for those who have a reaction and then a term for those who don't have a time. You can use this linear predictor in some parametric survival model, e.g. a Cox regression.
If all the subjects actually have a reaction to drug A but we just only observe a lower limit of this time for some patient, we have censored covariate. This article introduces a survival model with a censored covariate:
It might be useful to you.
To answer the question of whether or not the time to reaction on drug A can be used as a predictor in time to reaction to drug B, you can fit your model with and without the terms corresponding to reaction time on drug A and use e.g. cross validation to estimate expected prediction error for both models.