They say that right now there are about 12 million bitcoins in circulation or having been stashed away, and eventually the production of new coins will stop, so only around 21 million of them will be available, and no more... Now, what will happen next if it ever comes to that, and Bitcoin is finally accepted as a legal tender?

I think that nothing life-changing is actually going to happen. We have already been there. And by there I mean a time period in the 19th century, commonly referred to as a Free Banking Era when banks could issue bank notes against specie (gold and silver coins). It is during these times when the term inflation began entering into widespread usage and emerging in literature, though not as a reference to price changes but as something pointing to disproportions between paper representing money and the amount of specie actually available in the bank

So, this time instead of gold we will have Bitcoin (which will be hoarded as per Gresham's law) and all kinds of "paper" derivatives inevitably entering the circulation as a means of exchange. These "papers" allegedly backed up by Bitcoin will in fact leave behind them only inflation, even despite Bitcoin intrinsic deflationary nature...

And welcome back to fiat!