FWIW, I've been experiencing exactly the same issue. I bought an iPhone 6S 128Gb directly from the Apple store on Friday morning.
I had decided to start from fresh new phone rather than restore from the backup of my iP6, because I had the day off and it's fun to go through it from the start once in a while.
Anyway, there's definitely a problem with lag / unresponsiveness when launching it with the TouchID Home button. I'm not sure I noticed this much before I updated (OTA) to iOS 9.0.1, but that might be just because it was one of the first things I did... Nevertheless, it has been a problem since.
I now have the phone all set up with my apps etc reinstalled from scrath manually. I'm not going through that again any time soon; life's too short.
Reading through the various posts above it seems this might be resolved with a complete restore via iTunes, so I decided to give that a go after making an encrypted backup to my PC. I kicked off the full 2GB download of iOS 9.0.1 on my PC in preparation.
However mid way through the evening, with my iP6s nowhere near my PC, the thing died. Nothing from the standby button, Home button, nothing - just a blank black screen... It wouldn't switch on with the standby button alone, but I managed to bring it back to life with a hard reset (Home + Standby for 5+ seconds). It all seems absolutely fine now, and best of all - no more lag!
So the conclusion (to me at least) is something wasn't configured quite right and this system shutdown and forced hard reset seems to have resolved matters for now. I'll hold off on the full iTunes restore for now, and give it go for another day or so to see if the lag issue reappears. Fingers crossed!