Your power supply has two independent parts. One with inputs U1 and U2 (AC1 and 0V), and output on U9 and U10. The other with inputs U3 and U4 (AC2 and 0V), and outputs U7 and U8.
The toroidal transformer which will eventually power it has two separate secondary windings, which you will connect to the respective pairs of inputs. If you want to test it with another transformer (your wall wart), which has only one secondary output and hence only one pair of output connections, you will have to test the two parts of your circuit separately, one after the other.
But before you do that: You have not told uns what voltages your power supply is meant to produce, and your schematic does not state any voltage ratings for the capacitors. Can your circuit tolerate the 22V supply?
Also -- your power supply schematic does not show any fuses, on the primary as well as the secondary side of the transformer. You need fuses! Certainly for safe operation going forward; and they are also most helpful to protect the more expensive components in case you wired something wrong, and to protect yourself. Was this supply built from a kit, or from a building plan? What did that source have to say about fuses? What about other protective measures -- enclosure, safety ground?
As a general question: Are you sure that you are ready to build a mains-powered device from scratch? You could put not only yourself in danger while you are experimenting, but can also create shock and fire hazards for future users!