How long can you leave a badly injured hand untreated?

Why would you ? You should never leave a badly injured anything go untreated.
Depends. This depends on the injury and the degree of disability you can accept if untreated.Best recovery may require quick evaluation & rx.Delay may prevent re-establishing blood flow to starved tissues including muscles & nerves. If blood flow to all fingers & tissues is ok & you have normal sensation over the tissue you may be able to wait some.Keep it clean, elevated & seek care when possible.