The red light indicates that the battery is charging. What is the battery display in the phone showing? Have you fully charged the battery?
By replacing the battery you may have removed the 'registration' between the base and the handset.
Try resetting the handset to clear the registration information from the base and the handset.
Press & hold END and # at the same time until you see the System Reset menu.
If you still have the base the handset is registered to, select Deregister HS, then select the handset you want to reset. To confirm, select Yes. The handset displays Handset not registered
If you see a “not registered” message on a handset, you must register it to a base before using it.
Place the handset in the base; the display should say Handset Registering.
Wait until the display says Registration Complete (about 30 seconds), then pick up the handset
and press TALK/FLASH. If you don’t hear a dial tone or the display says Registration Failed, charge the battery completely, then try again.
By the way your phone is probably a Uniden Dect 1680.