I know that the old remotes work with the new system, but do the new remotes work with the old system?
The Wii U gamepad (the one with the touch screen) and Wii U pro controller will not work on a Wii, and in fact do not work on the Wii mode the Wii U itself has.
All Wiimotes, including those with Wii Motion Plus (the only "new" remotes I'm aware of) will work with both Wii U and Wii systems.
There are no new "remotes" for Wii U. The Wiimotes sold for Wii U use are just Wiimotes, there is no updated Wii U version of the Wiimote. (Note there IS Wii Motion Plus, an updated version of Wiimote, but it is completely compatible for (and was design for) the Wii.)