"Can an abortion provider legally offer a private abortion to a woman up to birth in the uk?"
There are already existing, very limited, reasons for why an abortion can be carried out after 24 weeks, that hasn't changed. In that situation, everything is legal.
What has changed is that if a mother obtains an abortion for any other reason after 24 weeks (eg she just wants to do it, or maybe she doesn't want a daughter etc) then that is no longer a criminal act for her, but it is still illegal for anyone else to assist the mother in an abortion in that situation.
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"But if it's now legal to abort up to birth isn't that going to be offered to women?"
That is why there is concern about the "pills by post" scheme, women lying or being coerced to lie about their pregnancies.
There is also a concern, since it would still be illegal for anyone to assist, that it may give rise to 'backstreet abortions' as used to happen in this country until the Abortion Act 1967.
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Just as an aside, when I was born, it was illegal for my mum to have an abortion (it was also lawful to pay her less than a man). This country has changed so much for the better during my lifetime.
However, as a pp mentioned above this is not a move in the right direction and will open the right to abortions up to further debate and scrutiny.