OP, it is common for London raised DC to live in halls for their first years. Even without budget constraints many drift home in either their second or third years when a social circle has been built and work pressures start to build. Coming home after living away is a different dynamic and we found that neither us nor DC wanted to slip back into a parent child pattern.
I agree with talking to your DD, and with MrKlaw’s sensible posts. Both mine were very happy to leave without student debt and both content to live quite frugally. I think we may have felt quite differently if they had been big spenders, and had expected us to fund them. (I think we would have expected them to get jobs to pay for extras.) Instead they had a perfectly happy time living like, well, students.
Two things to consider. The first term is more expensive than the third, especially if the third is short and dominated by exams. Things like subscriptions and sports kit and big social events. Some Universities even have two twelve week terms and then a six week exam term. Then for us a cookery course before starting proved a really good investment. A student cookery book and practice beforehand would also work. So they know how to throw together what is left in the fridge with pasta, risotto rice and so on. DD was astounded at how much some or her peers, often with parents who struggled to support them, were spending on things like takeaways, taxis, coffees etc. Or buying train fares at the station rather than booking in advance. And how much was wasted by not shopping around for utilities, cleaning the flat at the end, or not returning the modem. A bit of parental mentoring at the start of the second year has gone further than extra money. Also think about going to charity shops for crockery etc. Cheaper than IKEA and it means your stuff is more distinctive. And give DC your old pans, and treat yourself to new, rather than the other way round. Not having loan repayments when starting work will give DC greater freedom in terms of where they live and where they build their careers.
In short if you and they agree that they want to be debt free, then you as a family agree how it can be achieved, without scrimping on core things.