When you turn the ignition key to start the engine � the circuits powered via the ignition switch should remain on, and the starter solenoid is energized.
Since you are loosing the dash lights and the solenoid does not �click in�, you may have a faulty ignition switch or loose/faulty wiring. It might be worth checking all the fuses, including the high ampage ones, normally located close to the battery + terminal.
I would also double check all electrical connections which where changed as a result of replacing the engine. It is highly likely that you have introduced this fault through not correctly re-making all the necessary connections, or unknowingly pulled a connector loose, or shorted a circuit � taking out a fuse.