7:30am to 6pm is 10.5 hrs
for many nurseries only the "core" hours 9am-3pm can be used with the free hours so you may be paying for 7:30-9am and 3pm-6:30pm.
£263.50pm is £3,162 per year for 51 days childcare of 10.5 hrs
On 39 of those days you are getting 6hrs "free" albeit with an £11 consumables fee. On the other 12 days you have to pay for the whole lot.
£3,162-(£11x39)= £2,733 to cover 301.5 hours per year that are unfunded.
That works out as £9.06 per hour of non-free childcare. That's quite expensive but not unusual depending on your area.
You might possibly be only being allowed to use 3 hours free per day in which case the £2,733 is covering 418.5 hours of unfunded childcare which is £6.53 per hour which is more normal.
There are different ways of doing the maths, the nursery will have its own policy.
You can only generally get the 15 hours actually free if you find a nursery who wil let you do it as 3 hours a day, 5 days a week, term time only. This is normally only offered by Council-run nurseries.
The nurseries simply cannot afford to give you the childcare actually free. It costs them more to provide the service than the government gives them. They would go bankrupt without this mathematical hoopla.