For Essex schools, the waiting lists are handled by the LA until the end of August (and beyond for schools which are not their own admissions authority).
If you have applied to a Southend school, Southend Pupil Access Team handle waiting lists for all schools until 5th September this year and continue to handle waiting lists for SHSB, SHSG, Cecil Jones, Shoeburyness and Futures after that date.
There isn't much point in phoning either LA to find out where your DC are on their lists for a week or two, as some people can be very slow in accepting / turning down places (and in Southend, we have 3 weeks to make up our minds in the first place). Bear in mind that late applications can move your DC down a list just as other people turning down places move them up.
Two years ago, DD didn't get any of our preferences and we decided only to stay on the waiting list for Shoeburyness and King John, but stopped phoning up when she stuck at about no.71 for KJS and actually moved down from 82 to 84 for Shoeburyness (she flunked the entrance exams and with the schools about 5 miles in opposite directions from here she obviously never was going to get a community place
(However, we put her on the Continuing Interest list for SHSG and she did get an in-year place in July of year 7 when 3 vacancies came up, so there is hope, if at first you don't succeed).