(I'm taking the assumptions that Moore's law exists at all, really does end in 2020, and implies that processing power will stagnate...after all, there are theoretical limitations on processing power. Even if we're not nearing those yet, we just might do that one day.)
How will this affect Bitcoin?
It will continue to produce blocks at 10 minute intervals. It's designed to do so with practically any amount of computing power. No harm done at all.
How will miners be affected by this?
Without substantially better hardware coming to market every few months, difficulty will level off and reflect the investment people have in the network (instead of that plus the progress of fast-developing ASICs and Moore's law, as we have now). The difficulty may even drop now and then, especially whenever the block reward halves, but as long as it doesn't do so too quickly, we'll continue to have block confirmations averaging under 15 minutes. The lifespan of mining hardware will become more relevant, so people will want more reliability in their ASICs. Mining will become a bit less speculative (because the future difficulty will not be quite so unknown) and market forces will drive the profit margins to be narrower than they are now, putting it further out of reach of people without super-low-cost energy. This might result in some centralization of mining power, but probably no worse than the current GHash.IO situation.
As mining becomes less important to the economy of Bitcoin, it will need to stand on its own legs as a currency, or fall flat. I think it will stand.