To re-iterate what has been said above, do not despair! I am 6' 3" also, and I am within the size limits for all RAF aircraft. I think that amongst the ejection seat aircraft, the Canberra is the most critical with a buttock - knee limit of 650 mm (the one exception to the 660 mm quoted by rockmav). However, I doubt if any new first tourists will be sent to the Canberra. Apart from this, the Hawk is probably the tightest squeeze, especially if you have stout legs, as your shins can rub on the bottom of the instrument panel. However, at 6' 3" you should be OK. And I can concur that there is no overall height limit, just limits on individual parts.
One other factor to consider is weight. There is a maximum weight for each type of ejection seat. Fortunately, it is often possible to sort this out if slightly overweight! If you wish to discuss precise numbers on this, send me a personal message and I will find out if there may be any potential problems.