Restoring an 10.9 TM backup onto an iMac running 10.6.8 My original hard drive has failed. Would not go beyond grey screen.

It is in course of being repaired with a new h.d. The Apple tech said the machine would be returned to me with mac o/s 10.6.8 installed (this being the latest version possible without passwords etc, apparently)

I have a Time Machine backup on a separate HD. This backup has the latest 10.9 o/s on it together with the latest Iphoto app. and several other apps I use.

My question is: EXACTLY what is the procedure for restoring my repaired machine to it's former state, (Applications, Settings, Stored information etc.)

I would appreciate a simple step by step answer to this point.
Can't give  you exact, as you haven't said how you connected to your Time Machine disk.

I am going to assume it's USB, and recommend simply holding down the Alt key at boot time, slecting your Time Machine drive, and following prompts...