Threshold ionization mass spectrometry (TIMS) is one of two methods envisioned in ITER to quantify the helium (He) fusion product in the exhaust pumping lines during plasma discharges. We present the first demonstration of another potential application of TIMS in a tokamak environment, namely, the analysis of deuterium (D) and He outgassing following a plasma discharge i.e. during the post-discharge. This method has been tested with sub-second temporal resolution in WEST during its first He plasma discharges in the so-called He changeover experimental campaign. The calibration method of TIMS using a D plasma discharge is presented while the uncertainties related to TIMS during rapid pressure variations, i.e. upon plasma breakdown and plasma termination, are discussed. The first results obtained with TIMS during consecutive D and He plasma discharges in the full tungsten (W) tokamak WEST are reported. It is found that the time evolutions for He and D outgassing in the post-discharge are markedly different. On one hand, He outgassing is instantaneous and decays within 60 s until the He signal gets below detection level. On the other hand, D outgassing can reach a maximum up to several tens of seconds after the termination of the plasma and this outgassing can last for about 10 min. These striking differences should be related to different retention and outgassing from WEST plasma facing components, presently constituted of actively-cooled ITER-like W units and inertially cooled W-coated graphite. Potential mechanisms at the origin of the different outgassing behavior for D and He in W plasma facing components are discussed in light of a systematic analysis of the He and D gas balance and a macroscopic rate equation modeling of the D outgassing from the divertor strike points.