If I were to mount a Nikon 1 lens on a full frame camera what would the focal length be? I know that using a crop frame sensor means that the focal length is effectively increased. Are the Nikon 1 lens lengths recorded as full frame equivalent or is the length adjusted for crop frame?
10mm is 10mm. Focal lengths of almost all interchangeable lens camera systems are expressed in the actual focal length of the lens.There are certain technical reasons why this is so, but the simplest is that a lens' focal length is defined as the distance from the film plane needed when the lens is focused at infinity to cast point light sources as a single point on the film plane. This doesn't change with regard to sensor size. What does change with regard to sensor size is the angle of view or field of view (FoV) that a lens of a specific focal length will include in the part of the image circle that falls on the sensor.
Your Nikon 1 camera has a crop factor of 2.7X. what this means is that the 10mm lens made for the Nikon 1 gives the same FoV as a 27mm lens would on a full frame camera. But the optics in the lens still have a focal length of 10mm.
For more about how focal length, sensor size, and FoV are related please see What is crop factor and how does it relate to focal length?
Focal length is never adjusted for crop factor, since it's a physical property of the lens. It doesn't change just because you put a bigger sensor behind the lens. So, a 10mm lens on your Nikon 1 would still have a 10mm focal length on full frame.
What would change, if a 10mm Nikon CX lens worked like a Nikon FX 10mm lens, would be the field of view. Since the CX format has a crop factor of 2.7, that would mean that the field of view of a 10mm lens on full frame would look like a 10/2.7 => 3.7mm lens would on your Nikon 1.
The main problem here, however, is that a CX lens doesn't work like an FX lens--the size of the image circle the lens projects is only big enough to cover a CX sensor, not an FX sensor. So, if you actually could mount one on an FX camera (which you probably couldn't, because it would have to sit right where the mirrorbox goes) you'd probably only see a circle of the image in a mostly black field, because the FX sensor is so much larger than the image circle the CX lens would project. So, you wouldn't really get the FoV of a wider lens.