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Hi, yes there is a way he can regularise his stay in the UK without having to leave.
If you and your partner have twins, he will be able to apply to remain in the UK under the access to a child rules which are part of the immigration rules.
If your partner is required to have access to a child whom is present and settled in the UK, he will need to prove this by either a court order or affidavits and other supporting evidence, he will be entitled to make an application to the UKBA.
If granted, he will initially be granted a 12 month visa. Thereafter, he will be able to apply for Indefinite Leave to Remain.
I would strongly suggest that he have a solicitor prepare and submit the application for him.
I hope this answers your question. If so, kindly click ACCEPT.
If you wish to discuss, please feel free to ask further questions. Kind regards
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