I can see nothing wrong with "I wish our flat were a bit bigger." It would make no sense to say "I wish I would play outside," because would is the conditional form of will.... will is about inclination (what you want or intend to do) - if the only thing wrong was that you didn't want or intend to play outside then you could simply change your mind and, hey presto, problem solved. What you want to say is that you wish that you had the ability to play outside (you can't do it because your throat hurts too much)... so you need the conditional form of "can," (could).