I've had the same problem -- the option to switch to Fleet channel using the drop-down menu in the chat box disappears, apparently after switching characters from Fed to KDF or vice versa without logging out of account.
I have discovered, however, that the /fleet command works just fine (i.e. "/fleet Hi everyone!" sends "Hi everyone!" to fellow fleet members), so you're still able to access the channel, albeit indirectly.
I've also had the same problem with hear fleet chat from the wrong fleet, e.g. seeing chat from Fed side after switching to KDF.
One odd thing about these is that I'm not sure if they're consistently happening; sometimes I'm able to hear and speak in the correct fleet channel without a problem, but I'm not sure if it's just because I logged out out of habit rather than just switching characters.
Sometimes you can flip your chat tabs around or switch zones to get it working again. Probably a lot of the same bug that affected the private queues, maybe they should just disable the chat too? Seems like a good move