Yeah, it can, as long as the circuit is completed the bell will chime. The transformer is there to keep 220 volt from an outside switch (doorbell button) from electrocuting the caller!!! It would be extremely unusual to have a 220 v chime anyway. I've never seen one...
There's a couple of ways. 220v to transformer. 12V pos to chime on to switch and back to chime and back to transformer. Or vice versa. They are not polarity sensitive AFAIK. As long as the circuit is completed.
You need to get yourself a multimeter and start checkin continuity If you get continuity from one point to another, you can start drawing a diagram (better than mine above obviously!!!) on what you find and then decide what wires are broken in the circuit.
Otherwise, get your arse to B&Q and buy yourself one of those wireless set ups and discard your old system!