Crikey. First-time camping with kids on a campsite seems ...um... brave and noble. Or possibly insane
As someone who has camped a lot in places distinctly colder and more remote than anywhere in the UK (-30 and bears kind of thing), I second all the comments about how cold it is to go camping in the UK.
i also second the comment about buying stuff on eBay - it really is full of almost-unused camping gear - i have never bought anything new!
Get some of those plastic crate things that people mentioned upthread. One contains camping stuff, one contains food, one contains plats/mugs/cutlery/thermoses/cooking/washing stuff, one contains waterproofs, wellies, hats, gloves, etc, and a fourth contains everyone's spare set of warm clothes. THen each person gets a small backpack for stuff like toiletries and favourite toys etc.
Packing list:
Bloody big tent. 7 or 8 berth for 5 people sounds reasonable. Tent floor thing. Tent pegs. Spare tent pegs. Mallet. Gaffer tape (the expensive stuff is better) for fixing tent leaks. Brush and pan for sweeping dirt out of the tent at every available opportunity.
Self-inflating foam mattresses (one each). THermarest are totally great, but expensive.
Sleeping bags and fleece liners (liners can be made of sewn-up fleece blankets). Possibly one liner in the bag and one over it, if you get cold. Polypropylene fleece won't compress the sleeping bag too much.
Pee bottle for the uninhibited who don't like getting out of bed at night - those 1L wide-neck plastic Nalgene ones are excellent. If temperatures are below minus 5, though, do remember to stick it down your sleeping bag, as frozen pee is quite difficult to dispose of in the morning.
Head torch for each person, tent lamp that can be hung from the ceiling (it can just be another head torch, but lamps do make it easier to see). Spare batteries.
Water bottle for each person, thermoses, big water container.
Plastic plates, mugs, bowls; cutlery; washing-up bowl, green scratchy sponge thingy, dishwashing liquid, antibacterial hand gel. Food preparation stuff as required, depending what you plan to eat. Camp stove with appropriate pots (Trangia ones are easiest to use though a bit inefficient) if necessary.
Waterproof jacket and trousers, waterproof hiking boots, wellies for everyone. Fleece or wool hat and gloves, fleeces, spare fleeces, thermal top and bottoms, spare thermals, thick socks. Softshell trousers are good as waterproofs - they're waterproof enough, and you can get away with wearing them the whole time as just normal trousers, over some thermals. Don't wear cotton. You will get cold when it gets wet.
Oh and chairs, table as required.
Wine. Chocolate. Gin.
Good luck!!!