I should add that if offered a wait list place you will be offered a bit of time to think. Two weeks in our experience. So movement won't happen just at the start of term as the trickle effect takes a while to work through. Its not just the money. Its quite a tough decision to switch back to the school you had really wanted when you have accepted your DC will be going somewhere else. One dad resorted to trying out both school journeys, including following the school bus.
From observation schools can end up with some unawarded exhibitions/scholarships, because they used them to try to tempt exceptional candidates, the type who probably had several offers. They may also have the additional terms fees from the person who gave notice. If you are being offered a very late place it might be worth asking if the school can give you a bit of financial help. The school really wont want to have you mull over the offer for two weeks and then have to start again, especially towards the end of the summer term and into the summer holidays.
Registrars have the difficult job of getting the numbers right. If they think they may have a place they need to fill, they will want you to remain interested. And in contrast if there is very little chance they wont want to feed your hopes. I would be tempted to carefully draft a reasonably consise email explaining that you are still very interested and why, as a parent, you think this is the right school for your child. (Try to cover things that wont have been picked up as part of the selection process, eg ease of journey, settle quickly as lots of friends are going, know the school because been on holiday camps there and fell in love, extra curricular etc the message being that your DC will settle quickly and engage fully and constructively in school life ) Say you recognise that there may be some movement before the start of term, explain your current position, and ask if they could call to let you know the chances of movement. You need to manage your DCs hopes.
IME, and we are now at the end of the school years, most Registrars are very helpful and impressively organised (bar the notoriously snooty lady at one West London Girls School). Selection at any age is pretty imprecise, and if your child is on the wait list they are almost certainly good enough, in the same way that if they had been offered a place straight off they are almost certainly good enough. (Schools we know have dropped a small number of kids at 13 or 16 because they are either struggling or not working, whilst plenty of wait list kids go on to become academic stars.) Be honest, and communicate, and good luck.
Looking back my two DC have had a reassessment (aged 3, the poor boy was so overwhelmed first time he refused to speak - his nursery sucessfully argued that this was an "intelligent" response and he was allowed to spend a couple of days in the classroom instead), an 11+ place at a very aspirational school, and a place via wait list. Each was pretty stressful, but they subsequently thrived in each of these schools. Just one round of UCAS left for me to offer support, though hopefully the AS exams DD is currently studying for will go so well that applications will go smoothly) and then I can completely disengage.