Honestly I think you will be absolutely fine. Where I live is much colder than the UK, and while we have facilities for tea making etc in our stable and I have never once used them. For me key is base layers, and to keep moving.
What to focus on depends upon you livery arrangements. If pony is on DIY, and presumably to be turned out with others, then you will need to check if there is a rota for poo picking etc, and as others have suggested make up feeds, haynets etc in advance. If she is on half livery it is relatively easy, as all you will need to do is fetch in to tack up and ride. Your daughter will probably be fine as she will be moving, but you may get cold watching her, so good gloves and socks will help, along with lined waterproof boots. Dress in layers so you can easily take them off and put them on as needed. Always have spare socks and gloves in the car in case they get wet. Where I am is much colder than UK and my winter stable dress is base layer with longjohns and long sleeved upper, t-shirt, jeans, one thin pair of socks and one thick, fleece, gilet, wind/waterproof jacket, gloves and hat. Even in the coldest weather I usually end up taking off the gloves, hat and outer jacket.
Good luck and I am sure that you are going to find the new place a great new start!