I have three of these units for my PC's and HTPC's:
http://www.newegg.com/Product/Product.aspx?Item=N82E16842101225They work very nicely, and work as you described you would like as long as you have the PC set to automatically turn on when power is applied (typically a BIOS setting).
In other words:
1) Power fails
2) If power is restored, nothing happens [actually you'll get console messages at both power fail and power restore points ... but nothing happens to the PC :-) ]
3) After 15 minutes +/- go by (this time is configurable in the UPS monitoring software -- or you can let it run until the battery reaches a specified % of remaining capacity) the server will be gracefully shut down ... after which the UPS will shut itself down.
4) When power is restored, the UPS turns itself back on and reapplies power to the PC => if the PC is set to turn on when power is applied, it will reboot.