His mother, as a child saying an Angel had come to her (instead of her much older boyfriend), would be kept in hospital until the birth with a provisional assumption of psychosis, possibly as a result of substance abuse, grooming and child sexual exploitation. Joseph wouldn't be allowed anywhere near her or the baby and his saying he would marry her wouldn't carry any weight with Social Services.
After the birth, she would be carefully monitored and if she maintained that her baby appeared by magic, she would likely be transferred to a Mother and Baby unit. Unfortunately, due to her circumstances of her age, being traumatised by the boyfriend and suspected drug abuse or serious mental illness, she would remain in the care system and be expected to prove her abilities as a mother in extremely artificial circumstances.
In the intervening period, Joseph would be arrested on suspicion of having sex with a child and only bailed on condition that he does not contact Mary under any circumstances. Eventually, a DNA test would be ordered. However, this would not dispel Social Services' concerns, particularly if he also confirmed that an Angel came to him in a dream to say he should take the baby on as his own and marry the child who is his mother.
If she managed to keep up a charade of 'oh, silly me, I was very upset and scared, but of course I know he isn't the son of God', she might, upon reaching the age of 18, be allocated a bedsit by the local authority. Social Services would, however, remain heavily involved and if they got wind of Joseph still being in the picture, would be very concerned about the safety of both mother and toddler.
If Mary fails to comply with their expectations, it would be possible that an emergency care order would be requested and the child taken into care. It is possible that, due to being so young, it could be decided by the Court that his best interests would be served by adoption.
Assuming that Mary manages to keep the Social Workers happy, as little Joshua reaches compulsory school age, his disappearance, along with Mary and Joseph (because he had a dream that the child was about to be murdered) would result in a very public appeal for any information concerning the three of them, especially if anybody got wind of their possible intention to leave the country. There would also be a Court Order requiring the immediate return of the baby into the care of Social Services.
If they were discovered in an adjoining country or returned at a later date, both parents would be arrested and the child would again, be taken into care.