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You can only get a visa to live and work in the UK as the parent of a British child if you are not in a relationship with the father of the child, so this is not an option. Further details can be found here:
For an unmarried partner visa a requirement is to evidence that the previous marriage has broken down - this can be difficult to do if the marriage has not been dissolved as it is highly likely to raise the question of whether or not the relationship has truly broken down.
If you can evidence that the previous marriage has permanently broken down and perhaps provide an explanation regarding the reason for not having divorced AND evidence at least 2 years co-habitation then you might qualify for the Unmarried Partner visa (same qualifying criteria as a Spouse visa). Showing 9 years co-habitation and having children together could of course be argued to demonstrate the breakdown of the other relationship (unless he spent half his time in the US of course)!