I have to replace the alarm panel (keypad) in my house with a more up to date one. The old one did not have a transformer and was powered by a separate PSU supply providing the 12v supply. The new panel has a built in transformer that should be connected to the mains.

Is it possible that I can bypass the built in transformer and simply use the existing PSU supply? If so, can I connect the PSU +/- to the AC connections on the board?

What Panel do you have? Sounds like the Old keypad may have been the Control Panel/Communicator, and Batteries with a wireless receiver.

Most Alarm Systems/Control Panels Have a transformer that charges a 12 volt dc battery, that battery provides power for The Panel, Detectors, siren, everything, even when AC fails. These type of Panels need 4 conductors to the Keypad(s).
Commonly:
Red B+
Black B-
Yellow Data
Green Data
One is data in, the other is data out.