My son has applied for and successfully got a british passport for his daughter so she is now classed as a british citizen.
Just for the record, she was always a British citizen - the passport doesn't change that, it just allows her to travel outside the US and enter the UK as a UK citizen
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His rent is 500 pound a month and his council tax is about 60 pound a month if I have read your thread correctly it would seem that although he just about has the 100 pounds a week or so for him and his wife he does not have enough to include his daughter as well.
The minimum they will need to prove is £105.95 per week for him and his wife, plus £62.33 per week for their daughter, so that's £729.21 per month after rent and council tax.
He has been trying to get third party support but having initially said that she would support him his manager has not been forthcoming with her financial documents.
Is there any reason why you or another family member cannot be his third party sponsor?
Usually it is a relative who acts as a third party financial sponsor - if fact, it says in the official Entry Clearance guidance that the third party sponsor needs to be a family member (MAA4 on this page:
http://www.ukvisas.gov.uk/en/ecg/maintenanceaccommodation).
The third party sponsor has to provide their original pay slips, their original personal bank statements and a letter of financial support, so I can see why his manager would be hesitant to hand all that personal, private information over to him!
the problem is this his wifes mother is making her homeless on 15th of june so time is running out for her to apply for and get the visa, if he was to apply as the sole sponsor what would his chances be and also would the fact that she overstayed her visitors visa slightly and the fact that she was refused entry into the untied kingdom in November affect there application.
Her overstay and refused entry on their own shouldn't prevent her from getting the visa, but it could delay the processing time by 2-3 weeks... so she'd probably be looking at least 6 weeks for the visa to be processed (possibly up to 3 months or so). There is an option to pay for settlement priority processing (processing in 15 working days), but it costs an extra $300 on top of the $1,337 visa fee and because she has an overstay and a refused entry, it would likely take an extra 3 weeks or so on top of those 15 working days.
The problem is that without third party sponsorship or any savings, the application will be refused on financial grounds. Going by the calculations of your son's rent, council tax and the minimum amount needed, he currently falls short by £250 a month, and if they can't show they have at least that much, the visa application will fail.