Thank-you for your question. I will address each of the options available to you in turn:
(i) Apply in-country for leave to remain as a fiance
Persons who are currently in the UK with temporary permission to stay in a different immigration category (for example, as a student), cannot switch into this category in-country. Since your girlfriend does not currently have any valid leave to remain under any category she too cannot therefore apply in-country to switch into this category.
(ii) Apply in-country for leave to remain as a spouse
Before you and your girlfriend can marry you will need to obtain a certificate of approval (COA). Previously, the Home Office would only issue a COA to an applicant who had current leave to enter or remain in the UK for more than six months, with at least three months of that leave remaining when the applicant made his or her COA application. However, following a decision of the Court of Appeal on 23/5/2007, applications for a COA can now be made by individuals who do not have valid leave to enter or remain (including those who have overstayed, such as your girlfriend). Where an application for a COA is made by an overstayer, the Home Office will write to the applicant asking that they submit further information in support of their application to enable the Home Office to be satisfied that the proposed marriage is genuine. If a COA is granted, notice to marry then has to be given to a ‘designated office'. Marriage can then take place 15 days later.
However, getting married will not enable your girlfriend (wife) to remain in the UK. A foreign national who enters the UK in a different immigration category (for example, as a student), is only allowed to switch into the category of spouse in-country if they were given a total of more than six months' permission to live here since their most recent admission to the UK. If your girlfriend arrived on a visit visa then she would have been given 6 months leave to remain in the UK. It is not therefore possible for her to switch from leave to remain as a visitor to leave to remain as a spouse.
(iii) In-country human rights application
Your girlfriend can make an in-country application for Discretionary Leave (outside the Immigration Rules) on the basis that to require her to return to India would constitute a violation of her right to respect for her family life in the UK as protected by Article 8 of the European Convention on Human Rights. The Home Office would also have to consider your rights. However, in my opinion any such application would have limited prospects of success. The Home Office will almost certainly take the view that your partner has always known that she has had no lawful basis to stay in the UK and therefore any family life that has been established has been established in full knowledge that she may have to return to India at some point in the future. Accordingly, and given also the relatively short duration of your relationship and the fact that the baby is not yet born, the Home Office is very likely to conclude that it is reasonable to expect your girlfriend to return to India and apply for entry clearance and therefore her removal from the UK will not violate her (or your) human rights. In order to overcome this line of reasoning you would have to show that it is not reasonable to expect her to return to India. Any such argument would be difficult to sustain without concrete evidence in support.
(iv) Out-of-country application for entry clearance
In my opinion your girlfriend is most likely to be able to regularise her position in the UK by leaving the UK and applying for entry clearance from abroad. From abroad she could apply for entry clearance as either a fiance or a spouse. Although previous overstaying ordinarily provides a ground for the Home Office to refuse entry clearance under rule 320(7B), this provision does not apply in family applications by virtue of rule 320(7C). Accordingly, her previous overstaying should not be a bar to her application being granted.
I am afraid that I am not in a position to comment on your girlfriend's entitlement to maternity care on the NHS.
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