The offer we have says Thurs 5 March and the hopeful offer would also be the same day for acceptance at the latest. I would expect that the schools then see how many they have had accepted out of the say 2x offers / places, and if they are not oversubscribed for places at that point then they would start on the waiting/reserve list.
So accepting or rejecting an offer quickly or on the last day possible won't make any difference to people on the waiting list that haven't had offers, and the school may not have to go to the waiting list anyway.
Having now spoken to admissions at a couple of schools, that isn't quite right. All responses to private schools have to be in by 4/3 or near that date so you have to make your choice by then. For this reason they go to the waitlist asap, and certainly don't wait till the closing date when you have to pay a deposit on one school or another. The school I spoke to said they constantly update the picture they have and as soon as rejections start coming in they reoffer if it looks like they will go under their capacity. So holding on to offers you don't want does affect other candidates chance of getting the school they want. They are trying to avoid situations where people go on changing schools after acceptance date. So it all needs to be done and dusted by then.
In state schools it works slightly differently because of continuing interest runs, but people do miss out as some parents hold a state and private place.