Pretty sure the RAF used to have a restriction on Fast Jets as they told me at Northolt I'd fail the medical due to the fact I am slightly over 6'2" and the Tornados are built for the average size male which is 5'11" with a tolerance of +/- 3 inches. So between 5'8" and 6'2" is fine.
Apparently I could have flown the bigger planes though - Hercs etc.
I did hear that airlines have a wider tolerance and have seen taller pilots at airports than myself.
Not sure about being too short though? but as above, it would prove an advantage on a tight flight deck.
Had a quick look for the average height for the UK and is states 177cm for men and 163.2cm for women