DB,
The QFE does not always change when the QNH does - all to do with rounding down to whole numbers. At Heathrow the elevation is 80ft AMSL, therefore the QFE is sometimes 2mb and sometimes 3mb lower than the QNH.
95% of commercial crews use the QNH nowadays, therefore airport elevation will show on the altimeter when the aircraft is at the airport. If a pilot requires the QFE then he will ask for it, but this is a very rare occurrence. Use of QFE is mainly confined to private aviation nowadays. The last bastion of the QFE was the military, but I believe that this has also changed. Perhaps Whipping Boy's SATCO can confirm.