2005 BMW 325i Fan

That is probably by design because it is likely wired to run the AC compressor when defrost or even heat are requested and the fans are required when the compressor is running.

No even if I don't turn the heater on it the fan starts to accelerate and sounds very loud but doesn't shut off until I turn the car off

Somebody is going to have to use a scan tool to analyze the sensor data to determine if it's a temp sensor of something triggering that.

I have nothing came up I changed the temp sensor and thermo is there any way to test the electric water pump is working

Do you have a fan problem or an overheating problem? Which is it?

It started with a overheating light coming on and the fan was on super fast I changed the temp sensor and thermo now after the car is warm the fan starts and goes faster and faster but no overheating light comes on I just don't no what it could be it's driving me nuts

The first step is to pressure test the cooling system for leaks. You likely have lost coolant and have air pockets now.

It's not losing coolant do you think it's the water pump

The only thing I THINK is that it should be tested thoroughly to determine what is going on. You aren't going to find a magic answer online. The car needs to be troubleshooted "on site" by someone that understands the system.