Hey Debbie,
Okay I saw the head hurting comment and had to chime in here

(by the way from one mom to another, Excedrin migraine is a great thing to own when starting this whole visa process..trust me)
First off, I'm going to super duper uber strongly advise
against you coming over on a 'visitor' entry. Get the visa's...fiancee or spouse but get em ! Here's why.....
You run the danger of being denied access to the country if immigration even suspects you're coming over for any other reason than a vacation. Given the fact that you have kids with you, a fair amount of luggage, and my gut feeling is you'll have to make some special arrangements given your daughters condition it's not worth the risk.
The other thing that's really important here (and it was my main concern) is you want to be able to get medical established for your daughter right away when you get here (I know it was my concern with Ariel) and if you're here as a visitor you'll still get treated in an emergency etc. but you can't exactly go down to a Doctors surgery and register with them right away. Given everything else you have to deal with you need that peace of mind. Soooooooo I'm going to advise you to get a fiance or spouse visa first.
You can go to the British Consulate website and get the forms
http://www.britainusa.com/visas/visas.asp You will need to apply for a visa for all three of you (you and your daughters). Now unless you have full legal custody of your girls, you will need to get legal permission from the father to take the children out of the country. Feel free to ask any questions you want about that, every parent here has a story and there's almost nothing we can't answer when it come's to that situation

Depending on where you live, you'll apply for your visa at your nearest consulate (either by mail or in person). Once you have your visa, you're all set. When you get here you'll be able to register the girls for school, get the medical care going for your daughter, etc without any hassles.
Once again, holler when you have questions, but the first place to start is in getting your visas, you and significant other will have to chose if you want to come over on a fiance visa or if you want to get married in the States first and apply for the spouse visa.
In all seriousness Debbie, if I were you I wouldn't come over without a visa.....it's not worth the risk given your situation. It wasn't in mine.
Red