Hi, thank you for your question. Given your wife has now moved out of the property and you are the main carer of the children you can write to the child benefit agency directly to inform them of the change of circumstances. You can see more information and contact details about reporting changes here: https://www.gov.uk/report-changes-child-benefit
If she contests this (which she has no ground to as the children no longer live with her) you can apply to court for a child arrangement order that they live with you (also known as a residence order) and this order will enable you to prove that the children reside with you.
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I think you should write to the Child Benefit Agency explaining in fully the circumstances and detailing what you do to meet the care and day to day needs of the children, and what your wife does in relation to this, as well as her limitations due to her disabilities and ask for it to be transferred to your name.
If she attempts to leave with the children you should consider immediately applying to court for a child arrangement order which can be done on form C100 and a £215 court fee to your local family court, and if there is a risk towards the children due to her disability then you can do this on an urgent basis.
As the mother has parental responsibility and your intention is to move 250 miles away, you would need to consult the mother before such a relocation and she can apply to court for an order prohibiting you moving away. In the circumstances you should consider applying urgently to court for a child arrangement order for them to live with you which will allow you to transfer the benefits to your sole name, but again you should inform the child benefit agency of this first as they may agree to transfer without any order in place.