couple of points.
1. bertsmobile has really good advice...he is particularly reliable with his experienced suggestions.
2. just because you have a new battery does not mean you have GOOD WIRING AND CONNECTIONS. test and inspect every single red and black cable (heavy) wire and connection...Pay attention to the starter solenoid (red) and the flywheel solenoid starter (also red), and also the black heavy wire going to chassis ground. It might be just be a loose or corroded connection or a bad ground. If you have a clamp meter, you can measure the current across the heavy red cable connected between the starter solenoid and the flywheel solenoid starter. If the amps are low, you know you have a wiring issue a bad connection somewhere or the flywheel solenoid starter is bad. This assumes the battery you installed it GOOD AND FULLY CHARGED - not RUN DOWN from repeatedly trying to start the mower with it.!! yes?
and finally but certainly not this....this is one of those universal size fits all shade tree mechanics methods to check and engine isn't borked...it will not tell you what the problem is, but it helps you eliminate many other things
remove all spark plugs...rotate flywheel BY HAND. Engine should rotate easily with no resistance. If it doesn't, you know you have something related to engine (valves, auto compression release, bent valves, valves stepped out or damaged, borked crank, seized or broken piston/piston rings, failed or borked PTO (is belt under tension all the time?, even power off?)
hopes this helps....start with the simple things and work up to the harder things...in most cases, especially if mower is old, it's going to be a simple thing...like wiring that has corroded or a starter solenoid that