IN short, you've got to ask for your situation to be taken in to account.
It's likely that they will try and reject the application on the grounds of his overstay in the first instance. You have to pre-empt this by evidencing your circumstances and why it was necessary for him to overstay. You have to show that the circumstances were exceptional and therefore a willful overstay was the only option. If the UKBA find that there were exceptional circumstances then the application will be allowed.
Make sure you both make statements giving details of his visa, his overstay, a chronology of events leading up to and after his visa expired. If you can submit evidence to show your diagnosis or stay in hospital (eg. birth certificate, perhaps a statement from the doctor or hospital at which you stayed) and other corroborating evidence (eg. correspondence. State that he simply had not option but to overstay and care for you because he feared for the well-being of you and his upborn child.
If there were no other persons who could provide care for you then state this as well.
If there is a lag of time between when you recovered and when he actually left the country then you are going to have to explain this as well.
IN this case I would suggest getting an immigration solicitor to check your application before you submit it to see that you have adequately explained your circumstance, provided sufficient evidence and have not missed anything out.
You can find Uk immigration solicitors through the following Law Society Website search engine:-
If they reject the application then you should appeal - applicant's get a much better hearing of their case and supporting evidence on appeal.
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