For CCHS, the operative date is 'the time of application' and yes, for an on-time application to be made, that must be by 3ist October at the latest.
Admissions-Policy-for-Entry-in-2026-Approved-Date-Revised-7-Feb-25.pdf https://share.google/6Dl3Pd58TVvYOswvM
Noting also this:
In the case of a recent change of address during the admissions process additional
evidence will be required. This may be in the form of: a UK photocard driving licence
(if applicable), council tax bill and two utility bills, one of which must be the electricity
bill showing actual usage...
...and that if a change of address is made after 31st October, which would improve the ranking of the application, then this address won't be used until after the school takes over the waiting list from Essex County Council at the beginning of September 2026 (for most other Essex schools, a change of address is permitted up to a date around November 21st, plus late amendments are used for ranking after national offer day.
One rather obvious thing to consider, is whether one wants to live within 3 miles of HBS, or within 12.5 miles of CCHS, should one decide that one wants to move at all? I'm not familiar with the former, but am with the latter and would imagine Hampstead Garden Suburb and its environs to be in general quite different. There is also the question of a primary school place for the rest of year 6, something which is in short supply in many bits of Essex, but may be easier to come by in London.
@GutsyBear did your DD also sit the CSSE exam? All postcodes SS0 to SS9 inc are in catchment for SHSG and WHSG, with bits of SS6 (Rayleigh area postcode) also IC for CCHS. You can check individual addresses on the 'Find a School' look-up on the Essex CC website.
WRT moving for HBS, presumably the school's website indicates whether it would be worth it if you coukd move in and provide the relevant evidence within the next 23 days. On the other hand, although HBS does have a priority area, iirc, since it was introduced, only c.30 / 40% of places have been allocated IC.