Could the problem be that the new unlock feature is not yet set up to work with multiple macs ?
I am using the watch to unlock three different Macs, that are compatible with unlock.
It is working very well with my two MacBook Pros, when I am at home and have only one Wi-Fi available (unless I moved too far away from the iPhone), but when I am in my office at work, where my two Macs are on different networks for different Time Capsules, I can only unlock the Mac that is on the same network as the iPhone.
Is the network card of your Mac compatible? It should be for a Mac mid 2013 or newer.
But check, if the Mac lists it as compatible in the System profile: https://discussions.apple.com/docs/DOC-10672
Enjoy the weekend and the sunshine!