You need a very basic OS, you choices are very much dictated by your Architecture.
Another major concern is if this will be an embedded system, or more general purpose. I don't know your specific situation, so I'm unable to narrow the scope.
I'm also not sure what your wireless requirements are, did you mean something that receives a wireless transmission and then re-transmits. Such as in a wireless repeater? Or something that connects to a wireless network, then creates a completely new network using the connected network as a gateway? Either way any of the wireless router distro's would suffice. Such as, Tomato, DD-WRT, or OpenWRT.
You are not really asking for an 'OS', your asking for a distro right?
As you might have heard, Linux could run on almost any arch. Compile GNU, and now you have GNU/Linux. In reality, it's much easier said than done. The distro concept, really avoids the headache. I'll assume your looking for an embedded device distro, not a general purpose. Any of the 'regular' distro's, would obviously need a custom minimal installation. Here is a small list, you should research thoroughly before attempting an installation.
I386 or X86_64
Debian, Ubuntu, Gentoo, Arch, Slitaz, tinycore, or rPath
Armel or Armhf
Debian, Ubuntu, Gentoo, or OpenWRT
Slitaz-Arm, is very experimental last I checked. Hopefully we can get a proper embedded Distro in the non-X86 Arena.
Other, MIPS ...etc
Debian, Gentoo, OpenWRT, ... Your choices are very limited.