Earth pretty much looked like this not so very long ago:
Welcome to Snowball Earth, some 650 million or more years ago. And this happened within the Goldilocks Zone!
Rogue Earth probably won't end up looking like a cue ball, simply because absent the Sun's influence, there won't be much weather. Whatever's in the atmosphere will rain or snow until Earth is far enough away that incoming solar energy no longer affects ocean currents and winds. Eventually, the surface will just be nut (and bolt!) freezing temperatures and rapidly diminishing amounts of incoming heat and light. Bad news for us.
Liquid water would likely persist in the oceans, meaning those buggers that live deep down won't even notice that us surface dwellers have turned into ice cubes.
How quickly depends on several factors:
- Where Earth is, at the time of ejection, with respect to the direction of the Sun's travel around the Galaxy;
- Which direction Earth gets ejected (this is very important, because if Earth is ejected in the wrong direction, it will just plummet into the Sun and your whole project will be moot)
- How fast Earth is travelling
Rogue planets can zip right along, and if Earth is positioned "behind" the Sun's direction of travel and gets ejected back the way it came and at speed, we could be waving bye-bye to the Sun pretty quickly! If we end up heading in the Sun's direction, perhaps we won't notice much difference?