There might be a short-term blip of excess births as we all get used to the new equilibrium. That is sort of how we ended up with an excess of old people today. But since the magic that doubled our life expectancy probably didn't also double our food, oil, and fresh water supplies, we'd soon see birthrates drop to maintain equilibrium.
But why settle for doubling? Why not make humans immortals? Oops! Now you have abolished children! Sell that Toys R' Us stock.
Or is the idea that our Philosopher's Stone will enable us all to live the healthy lives of twenty-year-olds up to age 150, at which point everything collapses at once and we die in one instant, like the wonderful one-hoss shay? Um, good luck with that.
In summary: changing the nature of life expectancy will change the nature of our demographic weirdness, but is unlikely to produce anything less complicated.