Douglas Carswell remains an MP until he is formally appointed to 'an office of profit under the Crown' - the only way an MP can resign. In his case, it will be to the office of Steward of the Manor of Northstead, appointed in rotation with the better known Steward of the Chiltern Hundreds, which is currently held by Patrick Mercer. Carswell has to formally apply and the Chancellor has to agree the appointment - he could refuse, but that hasn't happened since 1842.
Once that has happened - and Douglas gets a copy of the warrant, which is addressed to just plain 'Douglas Carswell' now - the Clacton seat then becomes vacant and the writ for the by election can be moved once the House is in session. By convention, this is done by the Chief Whip of the party that has lost the MP, but that is only a convention - the LibDems moved the writ for the Oldham by-election in 2010 and a Tory MP moved the writ when Ulster's Unionist MPs resigned en masse. Any MP can move the writ.
It is open to amendment with regard to the date at which the Speaker is exhorted to act, which also sets the date for the election itself and it is also open to being defeated by a vote. I believe (thanks to Anthony J Wells of Polling Report) that if the writ is defeated, then it cannot be moved again in the same parliament, which would leave the electors of Clacton without representation until May 2015.
The earliest the writ could be moved would be next week, once the House is back in session, but I would be surprised if the writ is moved to set a date within the conference period - no party would really want their conference press impact overshadowed by the braying presence of Farage. Even if the writ is moved next week, I'd expect a date in October.