Can I mount the McLaren wheel on my new Logitech wheel base?

No, it is not possible to put a Fanatec Steering Wheel on a Logitech Wheel Base. And a Logitech Steering wheel cannot be used on a Fanatec Wheel Base either.
The situation is similar with other manufacturers like Thrustmaster, Moza etc.
Their is no cross compatibility in between different brands for Steering Wheels and Wheel Bases.

Fanatec Steering Wheels only work on Fanatec Wheel Bases.
Fanatec Wheel bases can only be used with Fanatec Steering Wheels.
(Please verify the information of the two explicit products to ensure compatibility.)