Have you heard it rains a bit in Yorkshire?
They'll keep you warm without getting too sweaty.
Pick a comfortable pair and test them on plenty of training walks. Walking shoes will not suffice on this sort of terrain - you need ankle support.
A single pair of decent quality socks is preferable to two thinner pairs as they'll usually be warmer and more comfortable.
A day pack with around 25-litre capacity should be fine in summer.
It's good to have one thin pair with a thicker waterproof pair as backup.
No need to spend a lot of money on this, just pack something thick and warm.
MAP AND COMPASS
And more importantly, learn how to navigate with them.
TREKKING POLES (OPTIONAL)
Great for easing pressure on knees on ascents and descents.
If the route's wet and muddy they help keep water out of your boots.
A thin lightweight pair to pull over your trousers if it rains.
FLEECE OR WARM LAYER
If it's warm and dry you'll get away with one, but it's best to be fully prepared.
DRY BAGS OR BIN LINER
Essential for keeping the contents of your rucksack dry.
HEAD TORCH AND BATTERIES
Could be a lifesaver if you get delayed on the hills after dark.
MOBILE PHONE AND PORTABLE CHARGER
Great for navigating but shouldn't be exclusively relied on, and vital if you need to call for help.
FIRST AID KIT
A basic kit is fine but consider adding relevant extras like a thermal blanket/survival bag and blister protection.
You never know, the sun may come out!
FOOD AND DRINK
Two litres of water per day and possibly up to four litres on hot days. Plenty of high-energy food and snacks for a long day on the hill.