Yes, usually something like the end of the first week in January.
However... @Tea4216 , be aware that admissions policies can change. For your DS's year of entry the consultation period for changes (e.g. bringing forward the hard cut-off date for being in catchment - the OSA isn't keen on random earlier dates such as date of registration to sit the exam, if a challenge is made, but a challenge to e.g. the use of the CAF deadline probably wouldn't be successful) would need to take place for 6+ weeks between 1st October and 31st January when he is in year 5. As of now, the earliest that a major change that would affect him could be implemented from the 2027 intake. You need to keep an eye out for notice of a consultation on the school's website.
As for how soon you could move out of area, having been offered a place - merely moving further away, but still in a catchment postcode wouldn't be a problem, as the allocation is score based and (currently) no postcodes have precedence. Moving out of catchment, trickier - a place obtained fraudulently can be withdrawn even after the child has started at the school and given how insistent Latymer is on only admitting from within catchment, I can't imagine that they wouldn't be suspicious of a move as earlh as during the first term. But there are loads of in-catcment postcodes, so presumably it wouldn't be too difficult to find somewhere 'nicer' than where you are currently, but still within catchment?