Did someone install new electronics (powerline LAN systems, solar power converters etc) near you? Or perhaps some of your own electronics? A particularly cheaply designed HF ballast can do it.
Certain electronics can puke out tremendous loads of interference. Solar power converters can be completely 'clean' but one of the things you have to do to make them clean, is to run cables next to each other and not in a loop over the roof to avoid making them work as transmitter antennas. In general, if solar converters are the issue, usually the interference gets less after sunset. No guarantee, though.
The worst solar setups i've encountered would already start interfering about 300m away, and could interfere both with AM/medium wave broadcast as well as with digital VHF emergency services.
I checked - currently the MSF signal is still audible, checked on a web SDR/receiver:
http://websdr.ewi.utwente.nl:8901/So the issue is either electronics in your house. Turn off all breakers in your house, then hit the sync button on the clock. Does it work now? Flip on one half of your breakers. Still works? Turn on half of the other half of the breakers. No sync anymore? It's on one of the circuits you just turned on.
No sync, even with your own power shut off? Then the problem is dirty electronics somewhere else in your block of houses. You can call Ofcom for that, i think. Electronics shouldn't interfere with such a strong signal.
In theory you can modify a clock to give it a headphone output if you add a little AM detector and an amplifier so you can hear the interference. That way you can use the directional built in antenna to triangulate the source of the interference. If it turns out to be the neighbors and you have a good relationship with them, it might be a better idea to just help them solve the problem, rather than sending the authorities to them.
If you have a ham radio operator near you, they'd likely be delighted if they can help with the right receiver to help triangulate the source, and feel Very Important. Especially because we're all quite concerned with keeping the RF spectrum free of pollution. It's like what light pollution is to astronomers.