As part of the insurance survey we are doing, it turns out there are a few Lumley Insurance clients among us. That is a company overlooked until now.
Lumley have a website here, and a web page devoted to the February 22nd Christchurch earthquake here. As with other insurers it has a complaints procedure which is outlined here.
Specific policy related information is not available so it may depend on who you got your insurance through. Click on the name of the organisation providing your policy to go to the relevant page.
Unlike the large insurance companies, Lumley don't seem to have a FAQ-style claims guide (or at least not on-line). If you want to get an idea of some of the issues that can arise from earthquake claims, I would recommend reading either the AMI (here) or State (here) claims guides. Neither of them are going to have any influence with Lumley, however they might help you to frame any questions you want to ask.