The British Blue Shorthair is of a very reserved temperament and will only play on her terms to ensure her dignity at all times. Once bored of games, she will quickly turn her nose up and look the other way. She is usually quite quiet and calm, only becoming vocal to alert you to feeding times or her disgruntlement.
She will need to be very comfortable in your company before she will trust you enough to allow you to pet or stroke her, but once she has built up this trust, she will just about tolerate you doing so and will probably endure rather than enjoy the experience. However, she does like being in your company even though she may not want to sit on your lap. She is rather nosy too and will follow you around the house just to see what you are up to.
She will definitely be the dominant character in your household so as long as you can accept this, you and she will get along well.