A relay would be the wrong approach, because the voltage must drop typically below 2V to deenergize a 5V relay.
And then the <2V should cause crazy reaction of your AT89C52.


Simple use two shottky diodes to drive your circuit from the power supply or from the LDO after the battery, depending, which voltage was higher.
You can use an adjustable LDO and adjust it so, that the power supply voltage was always higher, if the mains voltage was applied.

If I be right, the AT89C52 was designed to work on voltages down to 4V.


Peter