Taiwan Semiconductor (TSMC) is one of the few chip fabrication companies in the world (especially when you omit the memory producers, etc.). Their customers include: AMD, NVIDIA, Qualcomm, Broadcom, and even a few Intel Atom processors have come out of their lines at one point. They will take money from just about anyone who wants a chip.
According to Bit-Tech, a few customers will even have access to 16nm before the end of the year.
The catch, which of course there is one, is that production runs will be very small. We would love to see a gigantic run of new AMD or NVIDIA GPUs based on 16nm but that will not be the case (and not just because Volcanic Islands and Maxwell are both 2Xnm products). The first customers, while otherwise anonymous, will be interested in mobile systems-on-a-chip (SoCs).
On the plus side, when future 1Xnm designs come out, TSMC's production could be reasonably caught up to make a smooth launch.
Intel, the current leader in the fabrication world, targeted a slightly smaller 14nm process and have already begun producing a few odds and ends at that level. Full production has not even really started yet.
Just so you can get an idea of the complexity we are dealing with: 16nm fabrication creates details that are just ~32 atoms in width.