How long is it safe to leave a broken arm unattended for?

10-14 days: If you know your arm is broken, you need to have it treated by casting or surgery in 10-14 days after that treatment is more difficult.

Depends: Fractures that come through the skin are treated like emergencies, to minimize the risk of infection. Closed fractures are best placed into a reasonable position & splinted to reduce risk of further soft tissue injury and for comfort. Most fractures are treated within a week or so, but occasionally after failure of non-op treatment, surgeries may take place later. It's harder, but it can be don.