If this is a standard formula, how come NATS route charges are higher?
As
eglnyt has hinted, the formula is a means of sharing out each ANSP's total costs between all of its users. The square root formula seems a bit odd, I agree. It has been in place since the early 1970s (total cost recovery from en-route charges was, however, only achieved in the late 70s as it was phased in from 1974 on). I always assumed that the square root formula gives an approximation to the potential passenger payload of an aircraft, although I've never tried to do any analysis to confirm this.
ps I think that NATS is remarkably cheap, given that it is the only ANSP which has been established as a separate trading entity for long enough for any implicit state subsidy to have worked its way out of the system.