Not a lot, these days!
If the matter is serious enough, then the criminal law comes into play. But in most cases it is likely to be hard to interest the police and harder still to secure a conviction.
Other cases can be referred to the local Standards Monitoring Officer - usually the senior legal officer in the district council. The most that can be done then is to name and shame, although the mere fact of an investigation could be seen as a mild sanction.
Of course there's every reason to make such failings public and ultimately rely on the electorate to adjudicate, although that does require sufficient candidates for a poll.
ADDITION: The above answer looked at what would happen to the councillor in question and what could be done about the situation. There is another point that should be considered. If such a situation comes to light, the council could find itself in difficulties through acting in a biased way. That could result in liabilities in civil law that would be damaging to the council. Other councillors should, therefore, exert what pressure they can to ensure that interests are properly declared so as to avoid laying the council open to possible damages or penalties.
