I got this working today. I bought an Asus RT-AC68P router. It works out of the box. I just connected my phone to the router's USB 3.0 port using a normal phone usb cable (same one used for charging, etc.).
All Asus routers have the same firmware, so this should work "out of the box" with any new-ish Asus router.
I am not impressed with the Asus setup procedure when using a USB 4G modem/phone for WAN, but I ultimately got it working. I prefer working with dd-wrt myself.
This functionality is supposed to work with dd-wrt too, but today I also purchased a BUFFALO AirStation AC 1750 DD-WRT Router (WZR-1750DHPD) with firmware "DD-WRT v24SP2-(8/12/14) std" and I could not get it to work. I will try a newer or different version of dd-wrt in the near future.
Here are some references for more reading on the topic:
DD-WRT Forum :: View topic - 4G USB Modem
"I can recommend Kong Mod to achieve this. Works right out of the box,
at leastt for me. I recently lost my wired internet connection and
used a 3G dongle as backup. The only thing I did was to plug in the
dongle, select '3G/4G' as Wan connection and filled in the APN.
Everything worked directly, no additional modules to load. Success or
failure is of course dependent on the dongle being supported."
DD-WRT Forum :: View topic - BS build 14144 3G/UTMS option
Mobile Broadband - DD-WRT Wiki
As of 04/04/09 we support UMTS 3G / 3.xG / 4G/ LTE and CDMA2000(EV-DO)
modems connected via the USB port or as mPCIe cards.
Of Modems and Men
DD-WRT Router Benefits - #5: Take Full Use of a 3G USB Modem or 4G MiFi Mobile Hotspot
Share network of Android mobile phone to the router (USB 3G tethering) | SmallNetBuilder Forums
This page contains screen shots of the Asus router firmware and some
minimal instructions for how to use a 4G phone as the WAN connection.