In this kind of situation; the computer has no effect on whether the starter runs or not. The click you hear when trying to start indicates that either the battery is weak or discharged; or that the starter relay is bad; or the battery posts and cable clamps need a thorough scraping with a tapered reamer type battery cable service tool; or the ground cable is not solidly bolted to a clean engine bolt (the ground cable cannot be connected to a bracket, or to any painted or non metallic object. It will only work if it goes to a bolt that threads directly into the engine). There also must be a ground strap between the engine and the firewall. This strap may have been removed when the engine was taken out; and then not reinstalled with the replacement engine.
Try attaching jumper cables from a known good battery to the battery in the car. If this makes the starter run; it proves that the problem is in the battery or cable clamps. If a jumper battery connected to the car battery does not make the starter run; the battery cables may be dirty or not tightly connected at the starter solenoid, or the starter solenoid on the replacement starter is bad.
Some other, less common possibilities are that the ignition switch or neutral safety switch is defective.