Although not an Altham descendant, I did recently create a Wikipedia article on the ships Anne and Little James which came out to Plymouth Colony in America from London in 1623. The Little James had as its captain Emmanuel Altham, a young man from Essexshire and it was interesting to write of the historic misfortunes of the Little James and also of its captain, who had a sad demise.
Included in the article are something of his Essex roots, his family and of their London connection.
The source for most of this is a very excellent book by UK author Nick Bunker, which may have a little more on the Althams. The book may be available where you are - its listed as a source in the Wikipedia article and its complete title is: Making Haste from Babylon - the Mayflower Pilgrims and Their World - A New History. Pub. 2010. Just in itself, the book is a fine study in British politics, the world as King James saw it, etc.. in the early 17th century and covers as much about that as it does about early America.