Er no, the buzzer doesn't buzz or the light doesn't flash if the trailer bulb is out or the socket connection is dodgy! It's also a legal requirement.
Your socket won't be tested at the MOT. You can have a buzzer or light: they're both equally useless. The MOT tester won't even notice. Nobody cares.
Forgot to mention that recent legislation requires you to install towbar wiring while wearing only your pants. It's the law.
The relay senses the load of the bulbs. If there's less than three bulbs-worth, it won't flash the warning light.
You can fit a buzzer type relay in the engine bay. They go inline between the indicator wire and the 12N towbar socket. The "Ring RCT100" looks suitable. This would not require any change to the relay in the front fusebox. It just buzzes in the engine bay while indicating with something plugged into the towbar socket. You may or may not actually hear it.... http://www.ringautomotive.co.uk/files/mydocs/RCT100 Towing Kit Instructions LATEST.pdf You can also get ones which have fog light inhibit (so the vehicle rear fog lights don't light when towing, but the trailer ones do). This stops glare reflecting back at you from your trailer/caravan. The above relay does not have this feature.
Just wired my towbar by directly connecting to the spades at the back of light clusters and it's all working perfectly