The air filter removes impurity from the air that will be pulled in to the combustion chamber. If the air filter becomes wet, the water can seal off the filter and suffocate the engine. Remove the filter, it is usually a cylindrical plastic case with a foam core, and allow it dry out. Replace the filter and attempt to start the engine once more.
If the spark plug becomes wet, or if water gets trapped in the spark plug opening, it can cause misfiring and failure to start. Remove the spark plug and wipe it down. Keep the plug out for a few minutes and allow the port to dry out some before replacing the plug. Spray some starter fluid in the spark plug hole and replace the plug before attempting start the engine.
The small wire that attaches to the spark plug may have gotten wet, or its plug might be soaked. Remove the wire and allow it to swing free from the plug for a few minutes. Wipe off any excess water and replace the wire before again attempting to start the engine.
If the mower really got soaked, inner engine components such as the flywheel may have gotten wet. Some engines operate using magnets, and water can really put a damper on their functioning properly. This may necessitate allowing the mower to dry out for a few days before it will start. Allow at least a week before beginning to disassemble the mower to find the issue or returning it to the dealer for repair.