It sounds like the starter is engaging but not cranking the engine. Here are some things to check:
Starter Alignment: Ensure the new starter is properly installed and aligned.
Flywheel Condition: Inspect for damaged or missing teeth.
Battery and Grounds: Check all connections for corrosion and tightness, especially grounds.
Engine Seizure: Try turning the engine manually with a socket on the crankshaft pulley.
Wiring: Verify the starter signal wire and all connections.
Neutral Safety Switch: If automatic, try starting in neutral instead of park.
Since the issue started after the transmission replacement, double-check that no wires or grounds were left loose. Let us know what you find!