It would be worth getting this looked at by a solicitor or CAB, as it’s not a cut and dried case. You may have some legal cover on your home insurance or find someone who’ll assess your chances with a short free consultation.
You can’t assume the nursery could be sure you would vacate the place without you giving notice: they couldn’t rule out a deferred start at school, home schooling, or part time attendance.
OTOH you did give verbal notice. Whilst not ideal or 'normal practice' that might still be enough. Check the contract and other official communications. If the business has never said it requires written notice, then verbal notice from the customer may well be sufficient, leaving them to prove you didn’t tell them. Alternatively, if they acknowledged your verbal notice in writing that’s equivalent to written notice.
Even if it turns out you needed to give written notice, you may not be liable for the whole September payment. Under the Consumer Rights Act, customers have quite a high degree of protection when cancelling contracts.
The U.K. government website explains, "if you cancel the contract, the business is generally only entitled to keep or receive an amount sufficient to cover their actual losses that directly result from your cancellation." And, "businesses must take reasonable steps to reduce their losses (eg by re-selling the goods or services.)
Iow, they can’t sit back and take your money. If they can fill the vacancy from another family, they have no right to be paid twice (ie by you and the new customer.) Nor can they decide to bill you and not bother trying to fill the vacancy. So they must be able to prove they are actively taking reasonable steps to mitigate their loss: this means advertising vacancies and keeping a waiting list at all times. Instead of charging the full September bill, they probably should be charging you no more than an admin charge plus maybe the few days (if any) it takes to fill the space.
I would query the bill back to them. State that verbal notice was given and if they acknowledged this. If the contract doesn’t mention written notice, tell them you are confident that you fulfilled the contract. You might invite them to recalculate the bill, for your consideration, on the basis of clearly itemised charges for actual costs incurred in respect of administration or actual losses, after all reasonable measures to mitigate their losses. Request evidence that they have made every effort to fill the space. Inform them they may need to prove they are in full compliance with the Consumer Rights Act and you are considering reporting the case to your local Trading Standards Office.