Just updated my site from 2.3 to 2.4 and I had a similar problem than you.
If you have the correct login info and nothing happens when you input it at the login screen, then you might have an older version of php than the 5.5 version required by update 2.4.
The requirement change is clearly stated on the 2.4 update page but I had missed out that little detail. My site is hosted on godaddy and was running on php 5.2 if i recall correctly.
You might want to have a look on the control panel of your hosting/server provider...