It depends on the horse and to an extent the rider, an established competition horse with a rider that is also experienced then they could be out competing within days, the horse may well benefit from keeping to it's normal routine of working every day and getting on with the job, a green or young horse or less established rider will need longer to get a partnership so probably wouldn't get out so quickly and would probably benefit from going to a clinic or off site lesson before competing so they are setting themselves up to succeeed.
Most horses settle in to a new home best if they stay in a routine they are used to and at a similar level of work, I wouldn't give a competition horse time off as they are used to going out and about so are usually better kept in work from day 1.
As above, if the horse is reasonably experienced no reason not to get going straight away with a competent rider. If it is a bit more green then I would probably get out and about for lessons/arena hire before taking out competing.
There seems to be an idea that it is beneficial to give a new horse a couple of weeks off work to settle in a new place but in y experience it is better to keep them in a routine and in work.