Well it's not directly related to the size of textures. Textures in a game are going to be scaled in all sorts of ways depending on the surface they are on, how far away it is, etc.
Whether it's 720p or 1080p is really just about the amount of pixels it outputs. Upscaling something from 720p to 1080p means you are increasing the amount of pixels in some way, but not actually adding more detail/information eg. you might double some pixels up. If a console is really outputting say 720p, the graphics hardware is actually working out specifically what the colour values of all 1280x720 pixels should be.
Texture size only comes into it in an indirect way. There would be little point in outputting high res images when your textures look crap in the first place. For example if you grab an OpenGL port of Doom you can whack up the res as high as your comp will go but the game will still look like doom, just the edges will be less jagged.
I doubt console games use different textures for different resolutions. As they are a fixed spec, if it can run using the high resolution textures at a high resolution output, it will obviously be able to run the same high res textures at a lower output res. It might be a bit pointless if the output res isn't high enough to see much of their detail, but since your game is already running as fast as it needs to, there's no reason to use lower detail ones.
On a PC however, if your machine is struggling to run a game at a decent speed so you've put the resolution down quite low, you might as well use lower detail textures to give you a greater speed boost, since you probably wouldn't notice the difference anyway.