iPhone 5s starting up takes a lot of time I have an iPhone 5s and when the battery goes dead it takes a long time for it to become useful again after I plug it in. I timed this today and it took 15 minutes to become useful again. All my other phones (Android-based) never had this problem. 

My question now is, can I somehow shorten this time for it to start up?

Note: The iPhone is the property of my company, so jailbreaking it is not an option.
You can't short the time to use the device. It's normal you can't use your iPhone for at least 5 to 10 minutes untill it's 5%. It will auto startup the system and you can start using it again.