I have lots of little blackheads on my nose and I am looking for a new
facial cleanser (or facial cleansing regiment) that will help.

I am interested specifically in which products are best, since there
are a lot out there, and less interested in general advice about how
to deal with the problem.

If there is no one best facial cleanser for nose blackheads, I would
be interested in what kind of regiment would work best (say using this
product daily and this product once a week) - but again, specific
products please.

Have you looked at Bioré strips?

http://www.biore.com/usa/products/productInfo.asp?productId=10

A young friend of mine is wild about Clinique 7 Day Scrub Cream (Rinse
Off Formula). She uses it twice a week, and she says it has eliminated
her blackhead problem completely.

Speak to your pharmacist, as there are many products which are pharmacist-only.

I have been very pleased with Australian Bodycare cleansing wash,
which is a tea tree oil based preparation.