I'm the devil's advocate here, but , I'd say, don't worry that much.. After all, you are a member of that community and the library itself! You coincidentaly live very close, and it's not password protected.. I don't see how that could be a criminal offence under UK law, but I will check with my other half, who's a lawyer. Obvisouly this is as long as you don't engange in abusive behaviour or fraude of any sort online.. !
I think the problem will be to prove intent or absence thereof. I think the one who provides the service has a description of who it is for and under what circumstances. While I agree that the law doesn't have very strong provisions [yet!], I'm sure they're working on it, and a Google search reveals that people already get caught for wifi leeching.
The abstract problem is the following. When you leave the door to your apartment open or unlocked, does that give people you did not give explicit permission to enter a right to enter the premises? And even if you did lock the door, but the lock itself is cheap, does that give people you did not explicitly expect to enter a right to push the door open and enter the premises anyway? Or if we look at the library as a business: when you are expected to check in at the guard's counter, but the guard is not there for a second because he is in the restroom, do you still have a right to enter the premises freely? When you leech, you are using somebody else's resources for your own benefit in a way that was not intended by the service provider. As I said, it is VERY easy to prove whether you were physically in the library or not.
Of course, the leecher could always say: "I didn't know", but ignorance is often no excuse in the law. Hardly anybody will believe you got in somewhere "accidentally" for a period of several days, weeks, or months in a row. When you get caught, it is up to you to prove you didn't know. The law enforcers don't have to prove you did. These are cases where the *spirit* of the law may prevail, and not the letters or the words therein.
Paul