Chelmsford county high school for girls sets the CEM exam.
The other Essex grammars are covered by the CSSE exam. You apply to sit the exam in Spring, sit in September and hear the result in October. You apply for the grammars in the same way you apply for all other schools, you get six options, but it's wise to leave one space for a non-grammar school you are guaranteed a place at just in case. The minimum score last year was 301 for in catchment. 350 is what the most desirable schools ask for.
I think the forms have to be back in by January and then you hear in March.
Not all Essex grammars have in catchment and out of catchment scores. The Colchester super selectives and I believe one or two others have no catchments. They have a higher admitting score, but waiting lists can move up to twenty places before September and can go down up to twenty points lower than the stated admission score. Be realistic about journeys though, DD is at Colchester girls and some of her class come from East London and have on occasion fallen asleep in class. It's not an easy journey, even from Chelmsford.
If your kid needs to be tutored from Year 3 it's worth considering how they will manage in a high pressure fast paced environment. It's not easy and many of DDs friends are struggling, it's easy to feel like you are failing. Homework is intense and expectations are very very high.
DD needed six weeks of exam technique and gap filling before the exam, she is bright and motivated and was desperate to go the the grammar. The worst bit was waiting for the score and then paying for the uniform (ouch).
Another poster gave the link to a very useful forum, they have an Essex board. I think Redbridge has a separate board but I know very little about that area.