This tumblr post is the best explanation yet for why R2D2 wakes up late in The Force Awakens

The best explanation I’ve heard so far for why R2 only woke up at the end, is that he actually does start booting up when BB8 first finds him; he just has to get through 10 years worth of updates before that.

Or, of course, it could just be bad writing

Source: http://rowanwould.tumblr.com/post/136271291016/the-best-explanation-ive-heard-so-far-for-why-r2