I think I would go. It was pouring really heavily here yesterday and then as soon as the sun came out, the roads had rising steam and were dry very quickly. So the grass may dry out quicker if there is some sun between now and Thursday.
it's so much easier o get the tent pegs in the ground if it's been raining! Plus york has loads of indoor activities. The only thing I'd suggest is make sure you take some coat hangers for wet coats that you can hang near the entrance to the tent.