An HJM model is Gaussian HJM if vol term is deterministic. Note “vol” term means the coefficient of the dW term. Every Brownian motion must always refer to an implicit measure. In this case, the RN measure.
Well, Under the RN measure, the drift process is determined completely by the vol process. Both evolve with time, but are considered slow-moving  relative to the extremely fast-moving Brownian Motion of “dW”. Extremely because there’s no time-derivative of a BM
 I would say “quasi constant”