Moniker, what a fantastic, but daunting, position to be in! Many congratulations! Non-fiction isn't my area of expertise, but here are a few thoughts:
I would say (if it doesn't sound too bossy!
) your absolutely essential first step is to get in touch with the Society of Authors.
They are here:
http://www.societyofauthors.net/soa/section_page.php4?hp_nav_id=2&urlsection=About+the+Society
They have vast experience of all but the most minute publishers, will tell you what they know of the ones who are offering, advise you what questions you should be asking, and tell you what is - and isn't - acceptable in a contract. Once you've been offered a contract (or two, maybe, in your case?
) they'll go over it with their legendary fine-toothed legal comb and explain all the implications to you. They also have excellent booklets covering all sorts of issues - agents, tax, contracts, etc. etc..
You might want to consider getting an agent. They're not essential for non-fiction, as they virtually are for fiction, but if you're planning on a career in writing, they're worth their weight in gold. If a good agent takes you on as a client they're in it for the long term. They can negotiate better than most authors, not only for the size of the advance but also for what to expect in terms of publicity and marketing. They know what you can (and can't) expect of the publishing process of your book and can explain it, and are a big gun on your side to do the tough talking if necessary, so you can stay friends with your publisher. They may also be able to handle subsidiary (US, book club, audio book, translation, even movie?????
) rights for you, and the money you get will be up front, whereas with a publisher handling those it just goes to earn out your advance more quickly. In return they get 10% or more often 15% of any money that comes to you through a contract they handle.
The Society of Authors could advise you which agents to approach, and there's a full list in
The Writers and Artist's Yearbook, and
The Writer's Handbook. Good agents have very full books, but you're in the perfect position now, with so much interest. Publishers get pretty pissed off if they think they've got a contract agreed with an author, and the author brings an agent waltzing in to re-negotiate, but if the agent's in negotiations from the beginning it should be fine. It's not easy to recommend agencies, partly because it's a very personal business, finding an agent you get on with (VERY important that you do and they really 'get' your work in the way you do), and partly because some excellent agents are one-man-bands without a high profile. It's hard to judge how good they are from a listing, but major agencies you could approach from your (powerful) position include, and who would certainly have an agent who could handle this for you include: A P Watt, Aitken Alexander, Curtis Brown, Greane and Heaton, A M Heath, PFD (Peters, Fraser & Dunlop).
Very good luck with it all
Emma
<Added>Of agencies you could also try LAW - Lucas, Alexander, Whitley -or the big boys William Morris.