A competitor is using a name confusingly similar to mine, unregistered

You may be able to, even without registration, through passing off and the Fair Trading Act 1986. Passing off requires you to show that you have a reputation or goodwill in the name, that the competitor has misrepresented, deliberately or not, that its goods or services are yours or connected to you and that you have suffered or are likely to suffer damage. Evidence is important, including years of trading, sales figures, advertising and examples of actual confusion, such as misdirected calls or emails. The Fair Trading Act also prohibits misleading and deceptive conduct in trade. Send a letter setting out your position and asking the competitor to stop. At the same time, consider filing a trade mark application to secure your rights going forward. It is more difficult and costly than relying on registered marks.