It looks like Continental applies the elite bonus immediately.

I earned by 12th dine for lunch on the 16th and only got 3 miles. I had my 13th dine for dinner on the 16th and got 5+5 for that meal. I had my 14th dine for lunch the next day and got 5+5 for that meal.

My 12th dine posted first (I think on the 19th) and my 13th and 14th dine posted the next day (I think on the 20th).

So based on my experience, the elite bonus kicks in at least on the next day with Continental.