I currently use a laptop power adapter for the cigarette lighter, which is only on when the car is. The NUC's BIOS is set to turn it on when the power comes on, but I power it off manually before turning off the car. This is not ideal.
There is an external battery solution for the NUC, and I bought it! http://www.mini-box.com/NUC-UPS But I haven't switched to it yet. The big upside is that it will gracefully shutdown my NUC after I power off the car. But if I use the cigarette lighter for power, it will lose the ability to turn itself on when power is present, because the batteries are always present. NUC UPS supports using a different power connector which can turn on the PC on ignition, but I haven't had it installed in my car as of yet as that's a little bit more work, and I'd need some help from someone who knows more about the car's electrical system.
As far as fast power-on, it's a pretty high end i5 NUC with an NVMe SSD. It boots pretty darn fast, and my software takes less than a second to load once the OS is up. The slowest part of boot is that I don't want my location history easily steal-able, so it's encrypted, and I have to key in the code to unlock the machine. (I'm looking at a security key or similar to replace this step in the future.)
As a note, I'm more or less specced out what's involved in a solar power setup on the roof of my car to independently power my computer all/most of the time when the car is off and otherwise charge a secondary battery off the car's inverter, but there's no reasonable or sane reason to do it. ...But I thought about it.