my battery light doesn't illuminate... hasn't for years

when i first start the car the secs shows around 12volts with the enigne running. after ONE rev to 3/4k revs te alternator kicks in and i get a healthy 13.7 volts...

the mot tester mentioned that the battery light not working causes this need for a rev to start the car charging...

any thoughts on this?

is the battery light just a push in bulb or is it not a simple as that ie soldered in place or something?

cheers!