Can I use a 3 way mixer tap to install my UltraStream under the sink?

No.
The UltraStream makes molecular hydrogen which is a gas and the gas needs to breathe out of the UltraStream. This is why we use our special vented faucet in our UltraStream Undersink Kit.
Our vented faucet does not turn off at the faucet as all other faucets do,  but rather turns the water off prior to the UltraStream which allow the UltraStream able to vent the hydrogen.
A 3 way mixer tap is a closed system, meaning it will not allow the molecular hydrogen to escape.  The hydrogen gas will thave no where to go and will build up in the UltraStream.