Erm. How long do you get in the UK on a visitor's visa? I thought it was 6 months? No mention of traveling backwards and forwards to comply with the immigration requirements in the petition?
Op, I am confused. Which visa have you been living on in the UK on since 2012? If you have traveling backwards and forwards as suggested up thread, then your petition is a bit misleading, as you and your husband (and children) clearly have been managing for the timeframes you are out of the country. Your petition makes it sound as though you have been here constantly as a doting mother....
I don't want to sound unsympathetic. At 3, my third child wasn't walking or talking, and the hoops we had to jump through to emigrate were substantial. I know lots of couples who take extreme care to determine which immigration hoops they need to jump through in order to be able to get married and live together in whichever country is applicable. Some of these take a good couple of years to get set up. It's no fun. But they are responsible enough to investigate and understand the legal process, and are prepared to live apart in separate countries to fulfil the requirements while the paperwork goes through.
The difficulty here is that it sounds as though you are being disingenuous, and trying pull on people's heartstrings in order to shortcut the rules. Lots of us have disabled children. A fair amount of us are immigrants or emigrants from the UK. And most of us are sensible enough to understand the rules and follow them, or at least admit that we have fucked up. I'm not even sure if you have been in the country legally, as you are not really being open.
And having dealt with millions of bloody immigration lawyers in relation to our own case, I have never once had one suggest that I should just start a petition to emotionally blackmail the govt.
You are kind of right about the mn club aspect though - we have a great SN kids board that could have been a great support to you. It's a shame you didn't think to use it, instead of treating mn as an easily accessible service provider for petition names. You are still welcome. We're lovely.
Maybe tell us the real story, and we'll be able to offer some sensible advice that may be helpful. But don't just pitch up, want signatures, and get flouncy when you are asked some questions.
If it helps at all, I shortly have to apply for citizenship in my new country. There is a definitive chance that we will be refused (because of my youngest child's disability). We have always known this may happen, but have tried to ignore it. If it does, we are fucked. 
but I know the rules. We'll be selling the house, giving up our jobs, pulling the kids out of school and getting on a plane. We have been here for six years legally. I don't even want to think about trying to integrate three kids, including one with a disability, into a different school and health system.
Hopefully you will choose to share the reality behind the story (yes, yes, Betty's is lovely blah blah) and get some support.