This is something that's been annoying me for years now. Keyboard navigation used to work on Google with Firefox, but stopped a while ago. It still works on Chrome, and by now I'm pretty sure they're intentionally not fixing/enabling it on FF - not sure why though. I don't see any reason other than pushing users towards Chrome...
I find google.com is more broken on Chrome than in Firefox. In firefox, it just just annoying moves the search field as you start type. In chrome you are not allowed to use it at all. Instead it forces you to use the location bar.
Sometimes the location bar does a Google search. Sometimes it goes to the URL you type in. Sometimes, other things happen. Occasionally the thing that actually happens is the thing you wanted.
This only seems to apply to Chrome's default start page which looks like google.com, but actually isn't. It seems to work similar as in FF when navigating to google.com first.
Update: I just found out it's related to the "Google Instant" feature, which was somehow disabled in my case - it worked in private mode, but not when I was logged in. Enabling it brought back keyboard shortcuts.