How fussy is your circuit vis a vis the voltage regulation at 12V? One idea is to select a battery, e.g. 12V SLA which nominally float charges at 13.5V or so, and run it and your circuit in parallel. When the AC goes off, the battery will keep the supply between 13.5V and about 12.5V (assuming you don't completely discharge the battery). When AC is restored, assuming that the regulator is modified for suitable current-limiting, the battery will recharge at the same time the circuit remains powered. This is the way that automobiles and airplanes work...
If your circuit requires a more constant voltage, then "diode or" a battery with the output of the full-wave rectifier upstream of your voltage regulator. The battery voltage would have to be higher than the final output voltage+the dropout voltage of the reg+one diode drop, however. You will have to add a suitable charger for the battery, too.