I live in a rented flat that as two out-buildings on it, I have use of one of the out-buildings and another tents is to have use of the other, I have a bedroom directly above this out-building. Should a fire occur in this out-building and cause damage to my bedroom or other parts of the flat, will I be covered on my contents insurance.

I would double check with your agent, but I see no reason why it shouldn't. It doesn't matter how a fire started that caused damage (unless it was due to your negligence). The purpose of the insurance is to cover from accidental damage.