Expected wait time before departure Heathrow Express departs every 15 minutes. Assuming no bias when passenger arrives at the station, how much he should expect to wait before Heathrow Express begins its brief and expensive journey? Also, what is the right question: is it mean time before departure, median?

The time he should expect to wait is the mean of times before departure calculated using the probabilities of arriving at those times.

The time where he is as likely to have to wait more is as likely as having to wait less is the median time before departure.

Since the train leaves regularly every 15 minutes and the distribution of arrival times is uniform, both the mean and the median waiting times are (as Qiaochu Yuan says) 7.5 minutes.

That might be different in other situations: if the arrival or not of a train at a particular time did not depend on how long it had been since the previous train (a Poisson process with an exponential distribution of waiting times) but the average rate was one every 15 minutes, then the mean waiting time would become 15 minutes and the median waiting time about 10.4 minutes