Not much - but in a way, that’s a good thing!

The current vaccine is made for human flu variants. Avian H5N1 is currently pretty different from the circulating human variants that the vaccine targets. While this is why the human vaccine offers little protection against avian H5N1, it’s also why the likelihood of you catching avian H5N1 from a bird is relatively low.

The human flu viral variants have evolved to fit human cellular surface proteins, not bird ones, and the antibodies the vaccine generates are to the human viral variants. The avian flu variants have evolved to fit avian cellular surface proteins, which don’t look a lot like human ones.

As long as the avian virus is only fitting well to avian proteins, it will stay difficult for humans to catch it. Weirdly, if the human vaccine were to start offering us more protection - because the avian virus had evolved to start fitting human proteins better - we’d be in more trouble.