Ask the nursery for an itemised bill, detailing what all of the charges are for.
Note the following U.K. government advice, from their guide for consumers cancelling goods or services....
"Buseinesses can keep your deposit or advance payments, or ask you to pay a cancellation charge, only in certain circumstances. 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 (ie costs already incurred or loss of profit.) Businesses must take reasonable steps to reduce their losses(eg by re-selling the goods or services.) Non-refundable deposits should only be a small percentage of the total price. Cancellation charges must be a genuine estimate of the business' direct loss."
What you say about high demand suggests they should be well-placed to sell the place on. They cannot chose to just sit on the vacant place and take your money instead, as they have a legal duty to mitigate the loss, and a ready supply of families waiting for vacancies.
So, they should only be billing you for a fairly nominal administration charge.
It depends how you want to play it, but if it were me, I’d be offering the nursery manager one chance to get it right before I notify local trading standards officers.