What is the equivalent of an Australian or New Zealand honours thesis in British universities? Although it is somewhat discipline specific, in Australia, to receive honours for a bachelors degree, a student must undertake an additional, optional year of fulltime study, turning a three year degree into a four year degree. Generally, the vast majority of that year is spent conducting an independent research project which is written up as an honours thesis. An honours project generally has two supervisors, and is examined by two independent academics, including one from another university. In my field (human geography), they are usually 12,000 - 20,000 words long, and generally include the collection and analysis of new research data. Although they are somewhat more detailed and verbose, an honours thesis in my field should contain research equivalent to one or two academic papers in terms of the quantity and scope of the work, although obviously quality varies from student to student.

1st class honours in Australia is generally sufficient for admission to a PhD program here, although a scholarship will generally require either exceptionally high grades or additional demonstration of research potential.

I understand that a New Zealand honours is similar to its Australian counterpart.

I have supervised several honours theses in Australia. I am currently applying for lectureships in Britain, where I understand that the honours system is quite different. How can I explain succinctly to a British audience the scope and magnitude of honours supervision? Is there a simple equivalence with some other form of thesis in Britain?
in England, honours is granted after three years without a mandatory thesis or essay component. Scotland is more similar to Australia but not identical. 

Even in Australia, the requirement for an honours thesis varies. I don't see why you can't be explicit. Explain that honours is an optional fourth year similar to a one-year English masters and detail what fraction of the year was spent on the thesis. If your students' theses were publishable, get them published and say they led to publications.