Your safety and wellbeing is of primary concern. You must have the appropriate clothing to protect from cold weather conditions.
Layering can be a great method for keeping warm:
1. Under layer: A thermal underwear layer of long sleeved top and long johns bottoms are ideal. These are designed to wick sweat off the skin. Find examples on Amazon.
2. A normal t-shirt can be an extra layer if you feel the cold. Find on Amazon.
3. Base layer: A long sleeve fleece layer on top with high neck designed to retain warmth. Find on amazon.
4. Outer layer (shell layer): Shields you from wind and rain whilst stopping the cold penetrating. A hooded jacket is ideal. Find on Amazon.
Legs need protecting aswell. Warm thermal fleece lined trousers are ideal. Find on Amazon.
Around the neck I suggest a snood rather than a scarf. Find on Amazon.
Your head can be a large area where heat loss can ocure. A good wool hat with thermal lining or a Balaclava can be even better. Find on Amazon.
Your fingers are often the place where the cold is felt first and layering is often the best option here too. A thin inner or liner can work with a thicker outer layer. Ski gloves often have these options. The thin inner layer is great to keep the cold of the skin but allow you to feel the controls of your camera. Find on Amazon.
Lastly your feet. A good quality pair of waterproof breathable boots are needed together with good hiking socks.
These items can be expensive to buy and you may only use them on the one trip. So one great way to save is to rent your cold weather kit.
A high quailty waterproof down jacket costing around £500 to buy can be rented for around £60 per week. Check out https://www.outdoorhire.co.uk/ as an example of where to hire.