I emailed as soon as we received the waitlist to thank them and say the school was our top choice, how much we loved it and felt it was the right fit for ds etc. and we would accept the offer and pay the deposit immediately should we be lucky enough to get an offer. This was during covid so lots of people applied without seeing schools in person so I also mentioned we had visited twice.
Then the week before the state schools came out, I think the Wednesday/Thursday before, I emailed again just to say the school was still our top choice. In the letter they had said they offer on waitlist from a date 5 days after state results but we got a phone call the evening of the state schools results coming out offering a place. Good luck 🤞🏻